| L(s) = 1 | + 0.762·2-s + 3-s − 1.41·4-s + 2.42·5-s + 0.762·6-s − 7-s − 2.60·8-s + 9-s + 1.84·10-s − 0.750·11-s − 1.41·12-s + 2.96·13-s − 0.762·14-s + 2.42·15-s + 0.851·16-s − 2.55·17-s + 0.762·18-s − 1.81·19-s − 3.43·20-s − 21-s − 0.572·22-s + 0.658·23-s − 2.60·24-s + 0.876·25-s + 2.26·26-s + 27-s + 1.41·28-s + ⋯ |
| L(s) = 1 | + 0.539·2-s + 0.577·3-s − 0.709·4-s + 1.08·5-s + 0.311·6-s − 0.377·7-s − 0.921·8-s + 0.333·9-s + 0.584·10-s − 0.226·11-s − 0.409·12-s + 0.823·13-s − 0.203·14-s + 0.625·15-s + 0.212·16-s − 0.619·17-s + 0.179·18-s − 0.417·19-s − 0.769·20-s − 0.218·21-s − 0.121·22-s + 0.137·23-s − 0.531·24-s + 0.175·25-s + 0.443·26-s + 0.192·27-s + 0.268·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 0.762T + 2T^{2} \) |
| 5 | \( 1 - 2.42T + 5T^{2} \) |
| 11 | \( 1 + 0.750T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 0.658T + 23T^{2} \) |
| 29 | \( 1 + 7.52T + 29T^{2} \) |
| 31 | \( 1 + 4.45T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 9.95T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 + 5.06T + 53T^{2} \) |
| 59 | \( 1 + 9.40T + 59T^{2} \) |
| 61 | \( 1 - 4.39T + 61T^{2} \) |
| 67 | \( 1 - 4.88T + 67T^{2} \) |
| 71 | \( 1 + 2.98T + 71T^{2} \) |
| 73 | \( 1 - 0.752T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 7.76T + 89T^{2} \) |
| 97 | \( 1 - 6.08T + 97T^{2} \) |
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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
−7.45070705187939408569430865824, −6.63418966228066145742089417543, −5.90769308269100753518777084607, −5.49799692908375461674497063351, −4.61349772525928010456368769822, −3.84199912180274946761080983919, −3.25225584587613582020706849732, −2.31310071503741085598493628089, −1.50581895364491293653650860369, 0,
1.50581895364491293653650860369, 2.31310071503741085598493628089, 3.25225584587613582020706849732, 3.84199912180274946761080983919, 4.61349772525928010456368769822, 5.49799692908375461674497063351, 5.90769308269100753518777084607, 6.63418966228066145742089417543, 7.45070705187939408569430865824
