| L(s) = 1 | + 2-s − 0.423·3-s + 4-s − 0.487·5-s − 0.423·6-s + 2.40·7-s + 8-s − 2.82·9-s − 0.487·10-s + 1.38·11-s − 0.423·12-s − 1.29·13-s + 2.40·14-s + 0.206·15-s + 16-s − 2.88·17-s − 2.82·18-s + 0.654·19-s − 0.487·20-s − 1.01·21-s + 1.38·22-s − 8.21·23-s − 0.423·24-s − 4.76·25-s − 1.29·26-s + 2.46·27-s + 2.40·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.244·3-s + 0.5·4-s − 0.217·5-s − 0.172·6-s + 0.908·7-s + 0.353·8-s − 0.940·9-s − 0.154·10-s + 0.416·11-s − 0.122·12-s − 0.360·13-s + 0.642·14-s + 0.0532·15-s + 0.250·16-s − 0.699·17-s − 0.664·18-s + 0.150·19-s − 0.108·20-s − 0.222·21-s + 0.294·22-s − 1.71·23-s − 0.0864·24-s − 0.952·25-s − 0.254·26-s + 0.474·27-s + 0.454·28-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 | \( 1 + 0.423T + 3T^{2} \) |
| 5 | \( 1 + 0.487T + 5T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 - 0.654T + 19T^{2} \) |
| 23 | \( 1 + 8.21T + 23T^{2} \) |
| 29 | \( 1 - 4.28T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 8.21T + 37T^{2} \) |
| 41 | \( 1 + 0.313T + 41T^{2} \) |
| 43 | \( 1 - 4.40T + 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 - 8.99T + 53T^{2} \) |
| 59 | \( 1 - 0.393T + 59T^{2} \) |
| 61 | \( 1 - 8.49T + 61T^{2} \) |
| 67 | \( 1 - 3.95T + 67T^{2} \) |
| 71 | \( 1 - 0.756T + 71T^{2} \) |
| 73 | \( 1 + 9.00T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−8.076741522124953499147185083123, −7.26752049416809153588408493589, −6.49463806794834133691186801246, −5.64852503908849130534892238463, −5.19217584854963051736796970386, −4.22649415515331003343402890548, −3.64554006043013666246969600828, −2.44687583980629385921527231808, −1.69496731403412133200238194816, 0,
1.69496731403412133200238194816, 2.44687583980629385921527231808, 3.64554006043013666246969600828, 4.22649415515331003343402890548, 5.19217584854963051736796970386, 5.64852503908849130534892238463, 6.49463806794834133691186801246, 7.26752049416809153588408493589, 8.076741522124953499147185083123
