| L(s) = 1 | − 2.73·3-s − 5-s + 7-s + 4.47·9-s − 1.52·11-s + 3.26·13-s + 2.73·15-s − 2.63·17-s − 2.74·19-s − 2.73·21-s + 2.17·23-s + 25-s − 4.03·27-s − 3.93·29-s + 4.73·31-s + 4.16·33-s − 35-s − 9.94·37-s − 8.93·39-s + 1.30·41-s − 43-s − 4.47·45-s + 1.00·47-s + 49-s + 7.19·51-s + 4.69·53-s + 1.52·55-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 0.447·5-s + 0.377·7-s + 1.49·9-s − 0.459·11-s + 0.906·13-s + 0.705·15-s − 0.638·17-s − 0.628·19-s − 0.596·21-s + 0.453·23-s + 0.200·25-s − 0.776·27-s − 0.730·29-s + 0.850·31-s + 0.724·33-s − 0.169·35-s − 1.63·37-s − 1.43·39-s + 0.203·41-s − 0.152·43-s − 0.667·45-s + 0.145·47-s + 0.142·49-s + 1.00·51-s + 0.644·53-s + 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 11 | \( 1 + 1.52T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 + 9.94T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 47 | \( 1 - 1.00T + 47T^{2} \) |
| 53 | \( 1 - 4.69T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 - 1.74T + 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 - 5.23T + 89T^{2} \) |
| 97 | \( 1 + 9.16T + 97T^{2} \) |
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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
−7.58395026877266234196486865275, −6.82367505670753278740879433588, −6.30741206946002125407152921619, −5.54605716073890487622612218418, −4.93581514114343426314661355948, −4.28474444956939658808889892357, −3.43142063992428002218479159170, −2.11621690308810437949959828443, −1.02617105224621104663277151441, 0,
1.02617105224621104663277151441, 2.11621690308810437949959828443, 3.43142063992428002218479159170, 4.28474444956939658808889892357, 4.93581514114343426314661355948, 5.54605716073890487622612218418, 6.30741206946002125407152921619, 6.82367505670753278740879433588, 7.58395026877266234196486865275
