| L(s) = 1 | − 2-s + 4-s − 1.23·5-s − 8-s + 1.23·10-s − 11-s + 3.23·13-s + 16-s + 2.47·17-s + 7.23·19-s − 1.23·20-s + 22-s − 4·23-s − 3.47·25-s − 3.23·26-s − 4.47·29-s − 2·31-s − 32-s − 2.47·34-s + 6.94·37-s − 7.23·38-s + 1.23·40-s − 2.47·41-s − 10.4·43-s − 44-s + 4·46-s − 2·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.552·5-s − 0.353·8-s + 0.390·10-s − 0.301·11-s + 0.897·13-s + 0.250·16-s + 0.599·17-s + 1.66·19-s − 0.276·20-s + 0.213·22-s − 0.834·23-s − 0.694·25-s − 0.634·26-s − 0.830·29-s − 0.359·31-s − 0.176·32-s − 0.423·34-s + 1.14·37-s − 1.17·38-s + 0.195·40-s − 0.386·41-s − 1.59·43-s − 0.150·44-s + 0.589·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 0.763T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.50981064104359983985970421638, −6.87510090786233085536772419350, −5.92388809734210315233999904073, −5.52652317050472845038906012765, −4.50968305964953760476989565570, −3.55777187437865458875377950042, −3.16579956403131912652733825513, −1.95391792339217301259235669777, −1.13443543830627442846102815885, 0,
1.13443543830627442846102815885, 1.95391792339217301259235669777, 3.16579956403131912652733825513, 3.55777187437865458875377950042, 4.50968305964953760476989565570, 5.52652317050472845038906012765, 5.92388809734210315233999904073, 6.87510090786233085536772419350, 7.50981064104359983985970421638
