| L(s) = 1 | + 2-s + 4-s − 3.16·5-s + 8-s − 3.16·10-s − 11-s − 2·13-s + 16-s − 0.837·17-s + 1.16·19-s − 3.16·20-s − 22-s + 6.32·23-s + 5.00·25-s − 2·26-s + 4·29-s − 2.83·31-s + 32-s − 0.837·34-s − 4.32·37-s + 1.16·38-s − 3.16·40-s + 0.837·41-s + 6.32·43-s − 44-s + 6.32·46-s + 7.48·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.41·5-s + 0.353·8-s − 1.00·10-s − 0.301·11-s − 0.554·13-s + 0.250·16-s − 0.203·17-s + 0.266·19-s − 0.707·20-s − 0.213·22-s + 1.31·23-s + 1.00·25-s − 0.392·26-s + 0.742·29-s − 0.509·31-s + 0.176·32-s − 0.143·34-s − 0.710·37-s + 0.188·38-s − 0.500·40-s + 0.130·41-s + 0.964·43-s − 0.150·44-s + 0.932·46-s + 1.09·47-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 | \( 1 + 3.16T + 5T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 0.837T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 2.83T + 31T^{2} \) |
| 37 | \( 1 + 4.32T + 37T^{2} \) |
| 41 | \( 1 - 0.837T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 - 7.48T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8.32T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 + 1.16T + 83T^{2} \) |
| 89 | \( 1 + 3.67T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.40390696552429875805816987607, −6.78633192065727996855660760536, −5.88568236290735424417634580988, −5.07365129834296019028673126943, −4.55106686279006346469585476986, −3.87676902674504075288729074028, −3.13243545207358721453027083435, −2.51016169042402406257076724758, −1.19238049380801914993518122461, 0,
1.19238049380801914993518122461, 2.51016169042402406257076724758, 3.13243545207358721453027083435, 3.87676902674504075288729074028, 4.55106686279006346469585476986, 5.07365129834296019028673126943, 5.88568236290735424417634580988, 6.78633192065727996855660760536, 7.40390696552429875805816987607
