| L(s) = 1 | + 14·4-s + 91·16-s − 240·49-s + 32·61-s + 420·64-s + 1.56e3·109-s + 624·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 592·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 3.36e3·196-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 7/2·4-s + 5.68·16-s − 4.89·49-s + 0.524·61-s + 6.56·64-s + 14.3·109-s + 5.15·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.50·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s − 17.1·196-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(322.9571335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(322.9571335\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - 7 T^{2} + 7 p^{2} T^{4} - 7 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 60 T^{2} + 1542 T^{4} + 60 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 11 | \( ( 1 - 156 T^{2} + 31206 T^{4} - 156 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 13 | \( ( 1 - 148 T^{2} + 58438 T^{4} - 148 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 17 | \( ( 1 - 60 T^{2} + 18182 T^{4} - 60 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 19 | \( ( 1 - 500 T^{2} + 219142 T^{4} - 500 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 23 | \( ( 1 + 76 p T^{2} + 1319398 T^{4} + 76 p^{5} T^{6} + p^{8} T^{8} )^{4} \) |
| 29 | \( ( 1 + 2908 T^{2} + 3512038 T^{4} + 2908 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 31 | \( ( 1 - 2004 T^{2} + 2747046 T^{4} - 2004 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 37 | \( ( 1 - 2644 T^{2} + 4293766 T^{4} - 2644 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 41 | \( ( 1 + 2530 T^{2} + p^{4} T^{4} )^{8} \) |
| 43 | \( ( 1 + 3620 T^{2} + 8449702 T^{4} + 3620 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 47 | \( ( 1 + 3284 T^{2} + 12351526 T^{4} + 3284 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 53 | \( ( 1 - 7612 T^{2} + 27005158 T^{4} - 7612 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 59 | \( ( 1 + 2564 T^{2} + 9367206 T^{4} + 2564 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 61 | \( ( 1 - 4 T + 6406 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{8} \) |
| 67 | \( ( 1 - 3708 T^{2} + 2139558 T^{4} - 3708 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 71 | \( ( 1 - 17316 T^{2} + 124120326 T^{4} - 17316 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 73 | \( ( 1 - 17436 T^{2} + 129056006 T^{4} - 17436 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 79 | \( ( 1 - 8148 T^{2} + 89401638 T^{4} - 8148 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 2948 T^{2} + 15553318 T^{4} + 2948 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 89 | \( ( 1 + 2980 T^{2} + 82710022 T^{4} + 2980 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 97 | \( ( 1 - 17116 T^{2} + 216601926 T^{4} - 17116 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.26621948468211506259791855686, −2.25099710189270467750690215168, −2.16025465483065464938660565551, −2.11071331798509364631734171008, −2.03993564717152787707993895898, −2.00328408590846209386263181513, −1.91038813753261758842919008908, −1.90930137425510696094058703314, −1.70976757323669288963660894515, −1.67619753287759528951530976142, −1.65327678983586735165572585814, −1.61602409252877067594457253049, −1.57455819721233911568796578781, −1.26931532502127418199664839518, −1.25426044076713656301163480876, −1.17210765108381272646579039879, −0.879694144404883487115013433888, −0.838722909524131129365490169068, −0.791954348508190684516453742887, −0.70690757267547242432692705181, −0.52831083368867998296773564348, −0.49541075706872360937265106697, −0.33680399542935435997067257521, −0.31009917257067474484768750764, −0.28107617985868158022666400598,
0.28107617985868158022666400598, 0.31009917257067474484768750764, 0.33680399542935435997067257521, 0.49541075706872360937265106697, 0.52831083368867998296773564348, 0.70690757267547242432692705181, 0.791954348508190684516453742887, 0.838722909524131129365490169068, 0.879694144404883487115013433888, 1.17210765108381272646579039879, 1.25426044076713656301163480876, 1.26931532502127418199664839518, 1.57455819721233911568796578781, 1.61602409252877067594457253049, 1.65327678983586735165572585814, 1.67619753287759528951530976142, 1.70976757323669288963660894515, 1.90930137425510696094058703314, 1.91038813753261758842919008908, 2.00328408590846209386263181513, 2.03993564717152787707993895898, 2.11071331798509364631734171008, 2.16025465483065464938660565551, 2.25099710189270467750690215168, 2.26621948468211506259791855686
Plot not available for L-functions of degree greater than 10.
