| L(s) = 1 | − 2·5-s + 2·7-s − 4·13-s + 6·17-s + 2·23-s + 3·25-s + 6·29-s − 6·31-s − 4·35-s + 2·37-s − 6·41-s + 8·43-s + 8·47-s − 5·49-s − 10·53-s + 2·59-s + 4·61-s + 8·65-s − 14·67-s + 2·71-s − 12·73-s + 26·83-s − 12·85-s + 8·89-s − 8·91-s − 12·97-s + 30·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 1.10·13-s + 1.45·17-s + 0.417·23-s + 3/5·25-s + 1.11·29-s − 1.07·31-s − 0.676·35-s + 0.328·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 5/7·49-s − 1.37·53-s + 0.260·59-s + 0.512·61-s + 0.992·65-s − 1.71·67-s + 0.237·71-s − 1.40·73-s + 2.85·83-s − 1.30·85-s + 0.847·89-s − 0.838·91-s − 1.21·97-s + 2.98·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.932039050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.932039050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 104 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 125 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 177 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 119 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 176 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 26 T + 329 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 170 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75857196629752521578968267791, −7.73285072262981001995590787379, −7.29966232800536056885740354701, −7.24504605721799886520174161196, −6.52801297537257888775635143058, −6.42882046768721362081825343201, −5.71513830262932694941768005105, −5.61548937801420973302995396939, −5.07442331187307036500545587362, −4.83062437860869042103286902910, −4.48734213369371259375248458292, −4.21703479477678943176605880548, −3.54268788289314143687549297091, −3.40364813383620049691553412771, −2.85476452495387125961506831654, −2.61870125683197535643590804398, −1.74946996222356115703511041724, −1.71767752383627765032714147744, −0.77211104447975082383245136030, −0.55025575709203814693376252882,
0.55025575709203814693376252882, 0.77211104447975082383245136030, 1.71767752383627765032714147744, 1.74946996222356115703511041724, 2.61870125683197535643590804398, 2.85476452495387125961506831654, 3.40364813383620049691553412771, 3.54268788289314143687549297091, 4.21703479477678943176605880548, 4.48734213369371259375248458292, 4.83062437860869042103286902910, 5.07442331187307036500545587362, 5.61548937801420973302995396939, 5.71513830262932694941768005105, 6.42882046768721362081825343201, 6.52801297537257888775635143058, 7.24504605721799886520174161196, 7.29966232800536056885740354701, 7.73285072262981001995590787379, 7.75857196629752521578968267791
