| L(s) = 1 | − 2-s − 2.90·3-s + 4-s − 2.98·5-s + 2.90·6-s + 2.79·7-s − 8-s + 5.43·9-s + 2.98·10-s − 2.90·12-s − 6.11·13-s − 2.79·14-s + 8.66·15-s + 16-s + 5.07·17-s − 5.43·18-s + 19-s − 2.98·20-s − 8.11·21-s − 4.21·23-s + 2.90·24-s + 3.89·25-s + 6.11·26-s − 7.07·27-s + 2.79·28-s + 4.63·29-s − 8.66·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.67·3-s + 0.5·4-s − 1.33·5-s + 1.18·6-s + 1.05·7-s − 0.353·8-s + 1.81·9-s + 0.942·10-s − 0.838·12-s − 1.69·13-s − 0.747·14-s + 2.23·15-s + 0.250·16-s + 1.23·17-s − 1.28·18-s + 0.229·19-s − 0.666·20-s − 1.77·21-s − 0.879·23-s + 0.592·24-s + 0.778·25-s + 1.19·26-s − 1.36·27-s + 0.528·28-s + 0.861·29-s − 1.58·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 + 5.42T + 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 - 7.19T + 41T^{2} \) |
| 43 | \( 1 + 2.90T + 43T^{2} \) |
| 47 | \( 1 + 0.390T + 47T^{2} \) |
| 53 | \( 1 + 9.35T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 5.43T + 61T^{2} \) |
| 67 | \( 1 - 9.66T + 67T^{2} \) |
| 71 | \( 1 - 7.90T + 71T^{2} \) |
| 73 | \( 1 - 7.83T + 73T^{2} \) |
| 79 | \( 1 + 1.49T + 79T^{2} \) |
| 83 | \( 1 + 5.64T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.84826606521826930470606877830, −7.30320504104527173825785675476, −6.74767560711977933379860272740, −5.61659150808882489293694796501, −5.09974936680282697673088967580, −4.45434878896016815434229335584, −3.48879788847965042974854928909, −2.05082638278562736643140040683, −0.899158160604363313466553991207, 0,
0.899158160604363313466553991207, 2.05082638278562736643140040683, 3.48879788847965042974854928909, 4.45434878896016815434229335584, 5.09974936680282697673088967580, 5.61659150808882489293694796501, 6.74767560711977933379860272740, 7.30320504104527173825785675476, 7.84826606521826930470606877830
