| L(s) = 1 | + 2-s + 4-s + 8-s − 11-s + 4·13-s + 16-s + 3·17-s + 19-s − 22-s + 3·23-s − 5·25-s + 4·26-s + 9·29-s − 2·31-s + 32-s + 3·34-s − 7·37-s + 38-s − 6·41-s + 11·43-s − 44-s + 3·46-s − 3·47-s − 5·50-s + 4·52-s + 9·58-s + 9·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.301·11-s + 1.10·13-s + 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.213·22-s + 0.625·23-s − 25-s + 0.784·26-s + 1.67·29-s − 0.359·31-s + 0.176·32-s + 0.514·34-s − 1.15·37-s + 0.162·38-s − 0.937·41-s + 1.67·43-s − 0.150·44-s + 0.442·46-s − 0.437·47-s − 0.707·50-s + 0.554·52-s + 1.18·58-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.784428108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.784428108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.54103704229691802869687616440, −6.93172052689833324949088415584, −6.19230105399058659641664741935, −5.59571657494970378607076063962, −4.99631599606127517376670144416, −4.12700424451399342778163878757, −3.47942713570597072615942956836, −2.80948771527069125462220619710, −1.79827518816541420710486309331, −0.860696018107805749015849217116,
0.860696018107805749015849217116, 1.79827518816541420710486309331, 2.80948771527069125462220619710, 3.47942713570597072615942956836, 4.12700424451399342778163878757, 4.99631599606127517376670144416, 5.59571657494970378607076063962, 6.19230105399058659641664741935, 6.93172052689833324949088415584, 7.54103704229691802869687616440
