| L(s) = 1 | − 2·3-s + 4·5-s + 9-s + 3·11-s + 5·13-s − 8·15-s − 3·17-s − 2·19-s + 23-s + 11·25-s + 4·27-s + 6·29-s + 31-s − 6·33-s − 10·39-s + 5·41-s − 43-s + 4·45-s − 4·47-s − 7·49-s + 6·51-s + 5·53-s + 12·55-s + 4·57-s − 12·59-s − 2·61-s + 20·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1/3·9-s + 0.904·11-s + 1.38·13-s − 2.06·15-s − 0.727·17-s − 0.458·19-s + 0.208·23-s + 11/5·25-s + 0.769·27-s + 1.11·29-s + 0.179·31-s − 1.04·33-s − 1.60·39-s + 0.780·41-s − 0.152·43-s + 0.596·45-s − 0.583·47-s − 49-s + 0.840·51-s + 0.686·53-s + 1.61·55-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927831063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927831063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−9.028785550759821171663521683436, −8.249260173911303144493151953616, −6.68727230164453454450287089849, −6.41664595729362539794301783965, −5.94104483019529565461294896946, −5.12344995354976554150082178071, −4.32834845496254063722880661264, −3.01261032488267933122596519504, −1.83407206601526327017537809602, −0.986237026976612473156661495214,
0.986237026976612473156661495214, 1.83407206601526327017537809602, 3.01261032488267933122596519504, 4.32834845496254063722880661264, 5.12344995354976554150082178071, 5.94104483019529565461294896946, 6.41664595729362539794301783965, 6.68727230164453454450287089849, 8.249260173911303144493151953616, 9.028785550759821171663521683436
