| L(s) = 1 | + 2.43·3-s + 0.700·5-s + 2.92·9-s − 6.25·11-s − 3.33·13-s + 1.70·15-s + 17-s − 2.78·19-s − 5.97·23-s − 4.50·25-s − 0.194·27-s − 3.96·29-s − 1.55·31-s − 15.2·33-s − 2.53·37-s − 8.12·39-s − 5.62·41-s − 2.29·43-s + 2.04·45-s + 11.9·47-s + 2.43·51-s + 6.55·53-s − 4.37·55-s − 6.77·57-s + 8.78·59-s − 4.95·61-s − 2.33·65-s + ⋯ |
| L(s) = 1 | + 1.40·3-s + 0.313·5-s + 0.973·9-s − 1.88·11-s − 0.925·13-s + 0.440·15-s + 0.242·17-s − 0.638·19-s − 1.24·23-s − 0.901·25-s − 0.0373·27-s − 0.736·29-s − 0.279·31-s − 2.64·33-s − 0.416·37-s − 1.30·39-s − 0.877·41-s − 0.349·43-s + 0.304·45-s + 1.74·47-s + 0.340·51-s + 0.900·53-s − 0.590·55-s − 0.897·57-s + 1.14·59-s − 0.634·61-s − 0.290·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 - 0.700T + 5T^{2} \) |
| 11 | \( 1 + 6.25T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 19 | \( 1 + 2.78T + 19T^{2} \) |
| 23 | \( 1 + 5.97T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 2.53T + 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 + 2.29T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 6.55T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 + 4.95T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 8.40T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 - 4.93T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 0.898T + 97T^{2} \) |
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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
−8.224940103010341527205720737931, −7.67795787552456808348746575425, −7.12851256586622381230476649976, −5.83441652195539856130453164505, −5.25034617179900186399779572900, −4.18902966155811937794617474693, −3.35567479841298669818474194209, −2.32706642984166627661722289948, −2.11556989101790293111621871748, 0,
2.11556989101790293111621871748, 2.32706642984166627661722289948, 3.35567479841298669818474194209, 4.18902966155811937794617474693, 5.25034617179900186399779572900, 5.83441652195539856130453164505, 7.12851256586622381230476649976, 7.67795787552456808348746575425, 8.224940103010341527205720737931
