L(s) = 1 | − 1.96·2-s + 0.551·3-s + 1.86·4-s − 1.08·6-s − 1.56·7-s + 0.267·8-s − 2.69·9-s − 3.99·11-s + 1.02·12-s + 2.71·13-s + 3.08·14-s − 4.25·16-s + 7.60·17-s + 5.29·18-s + 0.708·19-s − 0.864·21-s + 7.84·22-s − 8.61·23-s + 0.147·24-s − 5.32·26-s − 3.14·27-s − 2.92·28-s − 5.83·29-s + 7.20·31-s + 7.82·32-s − 2.20·33-s − 14.9·34-s + ⋯ |
L(s) = 1 | − 1.38·2-s + 0.318·3-s + 0.931·4-s − 0.442·6-s − 0.592·7-s + 0.0947·8-s − 0.898·9-s − 1.20·11-s + 0.296·12-s + 0.751·13-s + 0.823·14-s − 1.06·16-s + 1.84·17-s + 1.24·18-s + 0.162·19-s − 0.188·21-s + 1.67·22-s − 1.79·23-s + 0.0301·24-s − 1.04·26-s − 0.604·27-s − 0.551·28-s − 1.08·29-s + 1.29·31-s + 1.38·32-s − 0.383·33-s − 2.56·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 3 | \( 1 - 0.551T + 3T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 - 0.708T + 19T^{2} \) |
| 23 | \( 1 + 8.61T + 23T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 - 7.20T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 9.24T + 41T^{2} \) |
| 43 | \( 1 + 0.217T + 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 1.31T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 2.79T + 67T^{2} \) |
| 71 | \( 1 + 3.67T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 - 7.53T + 79T^{2} \) |
| 83 | \( 1 - 5.42T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.893300608280413599484807258703, −7.64066306016155725224000359651, −6.16411141235641809041106558713, −6.00319899321138633799162011052, −4.92062776121757583181202056627, −3.79183350683684108230977723249, −2.97458211247408233184577154214, −2.22453296449998415709268054555, −1.03443964799426846646468603822, 0,
1.03443964799426846646468603822, 2.22453296449998415709268054555, 2.97458211247408233184577154214, 3.79183350683684108230977723249, 4.92062776121757583181202056627, 6.00319899321138633799162011052, 6.16411141235641809041106558713, 7.64066306016155725224000359651, 7.893300608280413599484807258703