| L(s) = 1 | − 2-s + 2.98·3-s + 4-s − 2.43·5-s − 2.98·6-s − 1.63·7-s − 8-s + 5.92·9-s + 2.43·10-s + 1.06·11-s + 2.98·12-s − 0.820·13-s + 1.63·14-s − 7.28·15-s + 16-s + 1.38·17-s − 5.92·18-s − 3.06·19-s − 2.43·20-s − 4.87·21-s − 1.06·22-s − 4.19·23-s − 2.98·24-s + 0.947·25-s + 0.820·26-s + 8.74·27-s − 1.63·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.72·3-s + 0.5·4-s − 1.09·5-s − 1.21·6-s − 0.617·7-s − 0.353·8-s + 1.97·9-s + 0.771·10-s + 0.321·11-s + 0.862·12-s − 0.227·13-s + 0.436·14-s − 1.88·15-s + 0.250·16-s + 0.336·17-s − 1.39·18-s − 0.702·19-s − 0.545·20-s − 1.06·21-s − 0.227·22-s − 0.875·23-s − 0.609·24-s + 0.189·25-s + 0.160·26-s + 1.68·27-s − 0.308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 2.98T + 3T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 13 | \( 1 + 0.820T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 - 7.38T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 + 7.95T + 43T^{2} \) |
| 47 | \( 1 + 1.00T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 - 3.91T + 59T^{2} \) |
| 61 | \( 1 + 5.48T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 1.77T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 0.0946T + 89T^{2} \) |
| 97 | \( 1 - 6.86T + 97T^{2} \) |
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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
−8.080800594760884282961851642963, −7.73673632272206663808920114030, −6.92771555319598608514726296018, −6.26558901198370107526523027335, −4.77249895638894455611366346438, −3.80053943174215678808231765818, −3.41380701332803609424029598302, −2.51165425490418456806058800246, −1.57538656921566313996786757926, 0,
1.57538656921566313996786757926, 2.51165425490418456806058800246, 3.41380701332803609424029598302, 3.80053943174215678808231765818, 4.77249895638894455611366346438, 6.26558901198370107526523027335, 6.92771555319598608514726296018, 7.73673632272206663808920114030, 8.080800594760884282961851642963
