| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s − 3·9-s + 4·11-s − 6·13-s − 14-s − 16-s + 2·17-s − 3·18-s + 4·19-s + 4·22-s + 23-s − 6·26-s + 28-s − 2·29-s − 4·31-s + 5·32-s + 2·34-s + 3·36-s + 2·37-s + 4·38-s − 6·41-s − 12·43-s − 4·44-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s − 9-s + 1.20·11-s − 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 0.917·19-s + 0.852·22-s + 0.208·23-s − 1.17·26-s + 0.188·28-s − 0.371·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s + 1/2·36-s + 0.328·37-s + 0.648·38-s − 0.937·41-s − 1.82·43-s − 0.603·44-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.543484966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.543484966\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.589845211470163293315170337629, −7.61743848507667801909991573698, −6.88347618540155716369999182925, −6.04379286865354323029440411987, −5.32653254961231840223487197286, −4.82596586255896887070784500201, −3.72445400208426810911395820935, −3.24912962756987522298833073827, −2.24035155691901429452468002749, −0.62039213891939089402340299147,
0.62039213891939089402340299147, 2.24035155691901429452468002749, 3.24912962756987522298833073827, 3.72445400208426810911395820935, 4.82596586255896887070784500201, 5.32653254961231840223487197286, 6.04379286865354323029440411987, 6.88347618540155716369999182925, 7.61743848507667801909991573698, 8.589845211470163293315170337629
