| L(s) = 1 | − 2-s + 4-s + 2.74·5-s − 8-s − 2.74·10-s − 11-s − 5.49·13-s + 16-s + 17-s + 8.03·19-s + 2.74·20-s + 22-s + 1.29·23-s + 2.54·25-s + 5.49·26-s − 4.54·29-s − 5.08·31-s − 32-s − 34-s − 4.03·37-s − 8.03·38-s − 2.74·40-s + 5.54·41-s − 4.03·43-s − 44-s − 1.29·46-s − 9.29·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.22·5-s − 0.353·8-s − 0.868·10-s − 0.301·11-s − 1.52·13-s + 0.250·16-s + 0.242·17-s + 1.84·19-s + 0.614·20-s + 0.213·22-s + 0.269·23-s + 0.508·25-s + 1.07·26-s − 0.843·29-s − 0.913·31-s − 0.176·32-s − 0.171·34-s − 0.663·37-s − 1.30·38-s − 0.434·40-s + 0.865·41-s − 0.615·43-s − 0.150·44-s − 0.190·46-s − 1.35·47-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 | \( 1 - 2.74T + 5T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 8.03T + 19T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 31 | \( 1 + 5.08T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 + 5.49T + 53T^{2} \) |
| 59 | \( 1 + 9.52T + 59T^{2} \) |
| 61 | \( 1 - 1.65T + 61T^{2} \) |
| 67 | \( 1 - 3.54T + 67T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 - 8.58T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 - 0.0872T + 97T^{2} \) |
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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
−7.52549781412875013337099053754, −6.77585713292461089085449624996, −5.99478621779306996641276815298, −5.25307533733522518018502556836, −4.97576629217043130456938403403, −3.54326963820004365151583007326, −2.80186771278151858170541182136, −2.04472049630453939729398549057, −1.31480939060367634484582915095, 0,
1.31480939060367634484582915095, 2.04472049630453939729398549057, 2.80186771278151858170541182136, 3.54326963820004365151583007326, 4.97576629217043130456938403403, 5.25307533733522518018502556836, 5.99478621779306996641276815298, 6.77585713292461089085449624996, 7.52549781412875013337099053754
