| L(s) = 1 | − 2.69·2-s + 5.24·4-s − 8.74·8-s + 5.89·11-s + 13.0·16-s − 15.8·22-s − 3.37·23-s − 5·25-s − 9.87·29-s − 17.6·32-s + 0.814·37-s − 3.34·43-s + 30.9·44-s + 9.09·46-s + 13.4·50-s − 2.03·53-s + 26.5·58-s + 21.3·64-s − 16.3·67-s − 14.1·71-s − 2.19·74-s − 7.42·79-s + 9.00·86-s − 51.5·88-s − 17.7·92-s − 26.2·100-s + 5.48·106-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.62·4-s − 3.09·8-s + 1.77·11-s + 3.25·16-s − 3.38·22-s − 0.704·23-s − 25-s − 1.83·29-s − 3.11·32-s + 0.133·37-s − 0.510·43-s + 4.66·44-s + 1.34·46-s + 1.90·50-s − 0.280·53-s + 3.48·58-s + 2.66·64-s − 1.99·67-s − 1.67·71-s − 0.254·74-s − 0.835·79-s + 0.970·86-s − 5.49·88-s − 1.84·92-s − 2.62·100-s + 0.533·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 5.89T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 + 9.87T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.814T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.03T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 16.3T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 7.42T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.532049309798470089052488358083, −7.64021378728423330942212927243, −7.16626128814891185361072003304, −6.27998907152896842068997125492, −5.80784063618549732365084195631, −4.17885337766834545879160965536, −3.28432283095661615451271831493, −1.99365323003417775263957010818, −1.37790193576724217795863225762, 0,
1.37790193576724217795863225762, 1.99365323003417775263957010818, 3.28432283095661615451271831493, 4.17885337766834545879160965536, 5.80784063618549732365084195631, 6.27998907152896842068997125492, 7.16626128814891185361072003304, 7.64021378728423330942212927243, 8.532049309798470089052488358083
