| L(s) = 1 | + 3-s − 2·4-s + 5-s − 7-s + 9-s − 3·11-s − 2·12-s + 15-s + 4·16-s − 17-s − 5·19-s − 2·20-s − 21-s + 25-s + 27-s + 2·28-s − 3·29-s + 10·31-s − 3·33-s − 35-s − 2·36-s − 2·37-s − 3·41-s − 43-s + 6·44-s + 45-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s + 0.258·15-s + 16-s − 0.242·17-s − 1.14·19-s − 0.447·20-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.557·29-s + 1.79·31-s − 0.522·33-s − 0.169·35-s − 1/3·36-s − 0.328·37-s − 0.468·41-s − 0.152·43-s + 0.904·44-s + 0.149·45-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.85526462667118, −12.68466177443134, −12.26616669527003, −11.51636339226174, −10.98105912032939, −10.36326931517569, −10.08864547717094, −9.818223125695775, −9.078096268463794, −8.840899242688375, −8.373548679098160, −7.934586132619154, −7.487720391812862, −6.777970817738065, −6.345471490788009, −5.822618628557301, −5.228375051058376, −4.829436233405659, −4.229420264361013, −3.889692567006094, −3.113751269345272, −2.738309620777320, −2.135902026980405, −1.478102554387648, −0.6725651291604327, 0,
0.6725651291604327, 1.478102554387648, 2.135902026980405, 2.738309620777320, 3.113751269345272, 3.889692567006094, 4.229420264361013, 4.829436233405659, 5.228375051058376, 5.822618628557301, 6.345471490788009, 6.777970817738065, 7.487720391812862, 7.934586132619154, 8.373548679098160, 8.840899242688375, 9.078096268463794, 9.818223125695775, 10.08864547717094, 10.36326931517569, 10.98105912032939, 11.51636339226174, 12.26616669527003, 12.68466177443134, 12.85526462667118
