L(s) = 1 | + 1.32·3-s − 1.90·5-s + 4.79·7-s − 1.24·9-s − 2.55·11-s + 0.665·13-s − 2.52·15-s + 5.36·17-s − 1.67·19-s + 6.35·21-s − 3.24·23-s − 1.36·25-s − 5.62·27-s − 8.72·29-s − 1.57·31-s − 3.39·33-s − 9.13·35-s − 7.50·37-s + 0.882·39-s + 10.1·41-s − 10.0·43-s + 2.36·45-s + 11.2·47-s + 15.9·49-s + 7.11·51-s + 11.5·53-s + 4.87·55-s + ⋯ |
L(s) = 1 | + 0.765·3-s − 0.852·5-s + 1.81·7-s − 0.413·9-s − 0.770·11-s + 0.184·13-s − 0.653·15-s + 1.30·17-s − 0.384·19-s + 1.38·21-s − 0.676·23-s − 0.272·25-s − 1.08·27-s − 1.62·29-s − 0.282·31-s − 0.590·33-s − 1.54·35-s − 1.23·37-s + 0.141·39-s + 1.57·41-s − 1.52·43-s + 0.352·45-s + 1.64·47-s + 2.28·49-s + 0.996·51-s + 1.58·53-s + 0.657·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 + 1.90T + 5T^{2} \) |
| 7 | \( 1 - 4.79T + 7T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 - 0.665T + 13T^{2} \) |
| 17 | \( 1 - 5.36T + 17T^{2} \) |
| 19 | \( 1 + 1.67T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 8.72T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 9.41T + 67T^{2} \) |
| 71 | \( 1 + 7.91T + 71T^{2} \) |
| 73 | \( 1 + 0.999T + 73T^{2} \) |
| 79 | \( 1 - 3.57T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 + 7.53T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62946196672178784093574332102, −7.34611642950015179047095571069, −5.67797748781099266447311521588, −5.58303892413956036953106568414, −4.49425427751683570006086842066, −3.91435406862648840925565521294, −3.14046289030797006363008509047, −2.19542003290671735817955356719, −1.47463457272959050984728300566, 0,
1.47463457272959050984728300566, 2.19542003290671735817955356719, 3.14046289030797006363008509047, 3.91435406862648840925565521294, 4.49425427751683570006086842066, 5.58303892413956036953106568414, 5.67797748781099266447311521588, 7.34611642950015179047095571069, 7.62946196672178784093574332102