| L(s) = 1 | + (0.698 − 0.449i)2-s + (−0.257 + 1.79i)3-s + (−0.544 + 1.19i)4-s + (0.959 − 0.281i)5-s + (0.624 + 1.36i)6-s + (−0.992 − 1.14i)7-s + (0.391 + 2.72i)8-s + (−0.266 − 0.0782i)9-s + (0.544 − 0.627i)10-s + (0.489 + 0.314i)11-s + (−1.99 − 1.28i)12-s + (1.92 − 2.22i)13-s + (−1.20 − 0.354i)14-s + (0.257 + 1.79i)15-s + (−0.219 − 0.253i)16-s + (−2.14 − 4.69i)17-s + ⋯ |
| L(s) = 1 | + (0.494 − 0.317i)2-s + (−0.148 + 1.03i)3-s + (−0.272 + 0.595i)4-s + (0.429 − 0.125i)5-s + (0.255 + 0.558i)6-s + (−0.375 − 0.432i)7-s + (0.138 + 0.962i)8-s + (−0.0888 − 0.0260i)9-s + (0.172 − 0.198i)10-s + (0.147 + 0.0949i)11-s + (−0.575 − 0.370i)12-s + (0.534 − 0.616i)13-s + (−0.322 − 0.0947i)14-s + (0.0665 + 0.462i)15-s + (−0.0548 − 0.0632i)16-s + (−0.520 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14559 + 0.487289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14559 + 0.487289i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (4.00 + 2.64i)T \) |
| good | 2 | \( 1 + (-0.698 + 0.449i)T + (0.830 - 1.81i)T^{2} \) |
| 3 | \( 1 + (0.257 - 1.79i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (0.992 + 1.14i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.489 - 0.314i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.92 + 2.22i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.14 + 4.69i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.88 + 4.13i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.778 - 1.70i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.31 - 9.15i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.36 - 0.401i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.71 + 1.67i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.212 - 1.47i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 6.45T + 47T^{2} \) |
| 53 | \( 1 + (3.92 + 4.52i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-5.66 + 6.53i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.17 - 8.20i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (6.20 - 3.98i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (10.5 - 6.76i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (1.38 - 3.03i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (2.37 - 2.74i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (1.54 + 0.455i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.72 - 11.9i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 4.11i)T + (81.6 - 52.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.55868648244002855828167418843, −12.84604664402763398203053185916, −11.59772064821042407871418234364, −10.60029976434398623129242009149, −9.604656629116885727284511174209, −8.578686340000528718999254587763, −7.01788909991388850316819424102, −5.24364645547849251895406550588, −4.31663942293678363214877363725, −3.04311605327226575506925067881,
1.70020396120515672489780080241, 4.10087123366448327620805419865, 6.08588147783356519364675952282, 6.18120350612378334190150444894, 7.72283758680449853406057101298, 9.230453429049502174620528933796, 10.20454786643484734617267745679, 11.64266776071365844960005050295, 12.76617584950136434445213575046, 13.41044036626572309731317263097
