| L(s) = 1 | − 3-s − 2·4-s + 7-s − 2·9-s + 3·11-s + 2·12-s − 5·13-s + 4·16-s + 3·17-s − 2·19-s − 21-s + 5·27-s − 2·28-s + 3·29-s − 4·31-s − 3·33-s + 4·36-s + 2·37-s + 5·39-s − 12·41-s − 10·43-s − 6·44-s − 9·47-s − 4·48-s + 49-s − 3·51-s + 10·52-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.577·12-s − 1.38·13-s + 16-s + 0.727·17-s − 0.458·19-s − 0.218·21-s + 0.962·27-s − 0.377·28-s + 0.557·29-s − 0.718·31-s − 0.522·33-s + 2/3·36-s + 0.328·37-s + 0.800·39-s − 1.87·41-s − 1.52·43-s − 0.904·44-s − 1.31·47-s − 0.577·48-s + 1/7·49-s − 0.420·51-s + 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.19963820545301, −13.56140659054097, −13.18862037522899, −12.42890223732650, −12.08583655901383, −11.75850135582913, −11.29639500781969, −10.34651292196137, −10.27380403568490, −9.635292975139011, −9.053388079919042, −8.644386183057064, −8.131935029046650, −7.615152881780787, −6.904604616471163, −6.432394138902663, −5.758181090492668, −5.251150589364819, −4.791283943606909, −4.465404331787488, −3.464549748582064, −3.266876481819237, −2.219970031646128, −1.531940943721233, −0.6746939193844037, 0,
0.6746939193844037, 1.531940943721233, 2.219970031646128, 3.266876481819237, 3.464549748582064, 4.465404331787488, 4.791283943606909, 5.251150589364819, 5.758181090492668, 6.432394138902663, 6.904604616471163, 7.615152881780787, 8.131935029046650, 8.644386183057064, 9.053388079919042, 9.635292975139011, 10.27380403568490, 10.34651292196137, 11.29639500781969, 11.75850135582913, 12.08583655901383, 12.42890223732650, 13.18862037522899, 13.56140659054097, 14.19963820545301
