| L(s) = 1 | − 3.41·3-s − 5-s + 4.24·7-s + 8.65·9-s − 3.41·11-s − 1.82·13-s + 3.41·15-s + 1.17·17-s + 2.58·19-s − 14.4·21-s − 4·25-s − 19.3·27-s + 5.82·29-s − 3.17·31-s + 11.6·33-s − 4.24·35-s − 5.65·37-s + 6.24·39-s + 1.82·41-s − 10.4·43-s − 8.65·45-s − 6.24·47-s + 10.9·49-s − 4·51-s + 5·53-s + 3.41·55-s − 8.82·57-s + ⋯ |
| L(s) = 1 | − 1.97·3-s − 0.447·5-s + 1.60·7-s + 2.88·9-s − 1.02·11-s − 0.507·13-s + 0.881·15-s + 0.284·17-s + 0.593·19-s − 3.16·21-s − 0.800·25-s − 3.71·27-s + 1.08·29-s − 0.569·31-s + 2.02·33-s − 0.717·35-s − 0.929·37-s + 0.999·39-s + 0.285·41-s − 1.59·43-s − 1.29·45-s − 0.910·47-s + 1.57·49-s − 0.560·51-s + 0.686·53-s + 0.460·55-s − 1.16·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 - 5T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 + 7T + 89T^{2} \) |
| 97 | \( 1 - 1.48T + 97T^{2} \) |
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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
−7.77710505130983978037427222543, −7.33372540589209196528732218477, −6.52582327214013996161603825823, −5.50551773325741218719591181767, −5.12035922100571593867202195668, −4.68326717101496282479031261263, −3.71905882575682884590083760968, −2.09503563778732560657747316865, −1.14796285619962485411544033116, 0,
1.14796285619962485411544033116, 2.09503563778732560657747316865, 3.71905882575682884590083760968, 4.68326717101496282479031261263, 5.12035922100571593867202195668, 5.50551773325741218719591181767, 6.52582327214013996161603825823, 7.33372540589209196528732218477, 7.77710505130983978037427222543
