| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 4.24·13-s + 16-s + 2.82·17-s − 4.24·19-s − 22-s − 6·23-s − 5·25-s − 4.24·26-s + 4·29-s − 7.07·31-s − 32-s − 2.82·34-s + 2·37-s + 4.24·38-s − 2.82·41-s + 10·43-s + 44-s + 6·46-s + 12.7·47-s + 5·50-s + 4.24·52-s − 2·53-s − 4·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.353·8-s + 0.301·11-s + 1.17·13-s + 0.250·16-s + 0.685·17-s − 0.973·19-s − 0.213·22-s − 1.25·23-s − 25-s − 0.832·26-s + 0.742·29-s − 1.27·31-s − 0.176·32-s − 0.485·34-s + 0.328·37-s + 0.688·38-s − 0.441·41-s + 1.52·43-s + 0.150·44-s + 0.884·46-s + 1.85·47-s + 0.707·50-s + 0.588·52-s − 0.274·53-s − 0.525·58-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 | \( 1 + 5T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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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
−7.61306098166435554070562429733, −6.65384527303565678162733941289, −5.96843709024867243502708045318, −5.67611566136597490879348210320, −4.30847609020242236869298969714, −3.87591841174532161644364408016, −2.91948815150303705673915894821, −1.96006000917457453118379825711, −1.22292499567795485421307076375, 0,
1.22292499567795485421307076375, 1.96006000917457453118379825711, 2.91948815150303705673915894821, 3.87591841174532161644364408016, 4.30847609020242236869298969714, 5.67611566136597490879348210320, 5.96843709024867243502708045318, 6.65384527303565678162733941289, 7.61306098166435554070562429733
