| L(s) = 1 | + 0.324·2-s + 1.44·3-s − 1.89·4-s + 0.468·6-s − 1.26·7-s − 1.26·8-s − 0.913·9-s + 3.90·11-s − 2.73·12-s + 2.19·13-s − 0.409·14-s + 3.38·16-s − 2.31·17-s − 0.296·18-s − 4.30·19-s − 1.82·21-s + 1.26·22-s + 0.680·23-s − 1.82·24-s + 0.713·26-s − 5.65·27-s + 2.39·28-s + 1.08·29-s + 8.23·31-s + 3.62·32-s + 5.63·33-s − 0.751·34-s + ⋯ |
| L(s) = 1 | + 0.229·2-s + 0.834·3-s − 0.947·4-s + 0.191·6-s − 0.477·7-s − 0.446·8-s − 0.304·9-s + 1.17·11-s − 0.790·12-s + 0.610·13-s − 0.109·14-s + 0.845·16-s − 0.562·17-s − 0.0697·18-s − 0.986·19-s − 0.398·21-s + 0.269·22-s + 0.141·23-s − 0.372·24-s + 0.139·26-s − 1.08·27-s + 0.452·28-s + 0.201·29-s + 1.47·31-s + 0.640·32-s + 0.980·33-s − 0.128·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.324T + 2T^{2} \) |
| 3 | \( 1 - 1.44T + 3T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 17 | \( 1 + 2.31T + 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 - 0.680T + 23T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 + 7.53T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 2.36T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 - 4.49T + 53T^{2} \) |
| 59 | \( 1 + 0.807T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 1.73T + 71T^{2} \) |
| 73 | \( 1 + 0.282T + 73T^{2} \) |
| 79 | \( 1 + 0.599T + 79T^{2} \) |
| 83 | \( 1 - 2.56T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 - 1.37T + 97T^{2} \) |
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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
−8.091492490430105461195920055352, −6.84112330544308955595362961200, −6.38651517017459642976749927714, −5.56215130711465270703164750559, −4.62743345873524920073499312688, −3.90330027134125544209537463138, −3.41937044116549531138897161593, −2.52483060603290627409267600389, −1.35624209125588488610863109453, 0,
1.35624209125588488610863109453, 2.52483060603290627409267600389, 3.41937044116549531138897161593, 3.90330027134125544209537463138, 4.62743345873524920073499312688, 5.56215130711465270703164750559, 6.38651517017459642976749927714, 6.84112330544308955595362961200, 8.091492490430105461195920055352
