| L(s) = 1 | − 2-s + 1.48·3-s + 4-s + 3.56·5-s − 1.48·6-s − 2.69·7-s − 8-s − 0.780·9-s − 3.56·10-s − 4.79·11-s + 1.48·12-s + 0.746·13-s + 2.69·14-s + 5.31·15-s + 16-s + 4.10·17-s + 0.780·18-s + 4.65·19-s + 3.56·20-s − 4.01·21-s + 4.79·22-s − 23-s − 1.48·24-s + 7.71·25-s − 0.746·26-s − 5.63·27-s − 2.69·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.860·3-s + 0.5·4-s + 1.59·5-s − 0.608·6-s − 1.01·7-s − 0.353·8-s − 0.260·9-s − 1.12·10-s − 1.44·11-s + 0.430·12-s + 0.207·13-s + 0.720·14-s + 1.37·15-s + 0.250·16-s + 0.994·17-s + 0.183·18-s + 1.06·19-s + 0.797·20-s − 0.876·21-s + 1.02·22-s − 0.208·23-s − 0.304·24-s + 1.54·25-s − 0.146·26-s − 1.08·27-s − 0.509·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 + 4.79T + 11T^{2} \) |
| 13 | \( 1 - 0.746T + 13T^{2} \) |
| 17 | \( 1 - 4.10T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 29 | \( 1 + 6.16T + 29T^{2} \) |
| 31 | \( 1 + 8.26T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 - 1.25T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 - 2.81T + 53T^{2} \) |
| 59 | \( 1 - 5.55T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 9.37T + 71T^{2} \) |
| 73 | \( 1 + 2.73T + 73T^{2} \) |
| 79 | \( 1 + 5.86T + 79T^{2} \) |
| 83 | \( 1 + 2.35T + 83T^{2} \) |
| 89 | \( 1 + 2.14T + 89T^{2} \) |
| 97 | \( 1 + 5.78T + 97T^{2} \) |
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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
−7.74594909579433598754967374594, −7.25253559071704180056084004788, −6.29140990074770562184514664137, −5.52375504798965815610609404754, −5.35214004820775784357527707965, −3.50344808802219484078498934956, −3.05448348791215272418115042995, −2.29987187469214904338629027113, −1.54416367102557312096080654637, 0,
1.54416367102557312096080654637, 2.29987187469214904338629027113, 3.05448348791215272418115042995, 3.50344808802219484078498934956, 5.35214004820775784357527707965, 5.52375504798965815610609404754, 6.29140990074770562184514664137, 7.25253559071704180056084004788, 7.74594909579433598754967374594
