| L(s) = 1 | − 0.445·2-s − 1.80·4-s + 0.246·5-s + 1.75·7-s + 1.69·8-s − 0.109·10-s − 5.65·11-s − 0.780·14-s + 2.85·16-s + 3.80·17-s + 5.58·19-s − 0.445·20-s + 2.51·22-s − 8.34·23-s − 4.93·25-s − 3.15·28-s + 5.93·29-s + 5.26·31-s − 4.65·32-s − 1.69·34-s + 0.432·35-s + 3.19·37-s − 2.48·38-s + 0.417·40-s − 0.445·41-s + 1.71·43-s + 10.1·44-s + ⋯ |
| L(s) = 1 | − 0.314·2-s − 0.900·4-s + 0.110·5-s + 0.662·7-s + 0.598·8-s − 0.0347·10-s − 1.70·11-s − 0.208·14-s + 0.712·16-s + 0.922·17-s + 1.28·19-s − 0.0995·20-s + 0.536·22-s − 1.74·23-s − 0.987·25-s − 0.596·28-s + 1.10·29-s + 0.946·31-s − 0.822·32-s − 0.290·34-s + 0.0731·35-s + 0.525·37-s − 0.403·38-s + 0.0660·40-s − 0.0695·41-s + 0.261·43-s + 1.53·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.105059687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.105059687\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.445T + 2T^{2} \) |
| 5 | \( 1 - 0.246T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 + 0.445T + 41T^{2} \) |
| 43 | \( 1 - 1.71T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 - 1.06T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 - 5.96T + 67T^{2} \) |
| 71 | \( 1 - 5.71T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 0.137T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−9.787368765939619880586732472523, −8.434419029577566170643483902832, −7.963600563906572109736585924715, −7.51820538255984659741110984937, −5.94519818367166696303144587104, −5.28529985354252275575319835876, −4.57701405624945159450179693223, −3.47772653255490280471366476835, −2.24469551193284360151582251701, −0.793846549474346026068604211073,
0.793846549474346026068604211073, 2.24469551193284360151582251701, 3.47772653255490280471366476835, 4.57701405624945159450179693223, 5.28529985354252275575319835876, 5.94519818367166696303144587104, 7.51820538255984659741110984937, 7.963600563906572109736585924715, 8.434419029577566170643483902832, 9.787368765939619880586732472523
