| L(s) = 1 | + 1.83·2-s − 0.379·3-s + 1.35·4-s − 0.695·6-s + 4.92·7-s − 1.18·8-s − 2.85·9-s − 5.22·11-s − 0.514·12-s + 4.04·13-s + 9.02·14-s − 4.87·16-s − 0.556·17-s − 5.23·18-s − 2.38·19-s − 1.87·21-s − 9.56·22-s − 9.05·23-s + 0.448·24-s + 7.41·26-s + 2.22·27-s + 6.67·28-s + 6.65·29-s + 6.74·31-s − 6.56·32-s + 1.98·33-s − 1.01·34-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.219·3-s + 0.677·4-s − 0.284·6-s + 1.86·7-s − 0.417·8-s − 0.951·9-s − 1.57·11-s − 0.148·12-s + 1.12·13-s + 2.41·14-s − 1.21·16-s − 0.135·17-s − 1.23·18-s − 0.547·19-s − 0.408·21-s − 2.04·22-s − 1.88·23-s + 0.0916·24-s + 1.45·26-s + 0.428·27-s + 1.26·28-s + 1.23·29-s + 1.21·31-s − 1.16·32-s + 0.345·33-s − 0.174·34-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 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 + 0.379T + 3T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 + 0.556T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 + 9.05T + 23T^{2} \) |
| 29 | \( 1 - 6.65T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 0.217T + 59T^{2} \) |
| 61 | \( 1 + 7.17T + 61T^{2} \) |
| 67 | \( 1 + 3.80T + 67T^{2} \) |
| 71 | \( 1 + 8.97T + 71T^{2} \) |
| 73 | \( 1 - 0.421T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 4.00T + 83T^{2} \) |
| 89 | \( 1 + 3.16T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.950049633522063532948443519160, −6.70054889791097460341968358835, −5.95914563465824429620632747809, −5.47065187171852265626949231215, −4.76641558524888361269449095587, −4.39451030146499853419207874399, −3.30251837697326845997519831879, −2.52910123987786757990432423413, −1.65466792678755051676265689276, 0,
1.65466792678755051676265689276, 2.52910123987786757990432423413, 3.30251837697326845997519831879, 4.39451030146499853419207874399, 4.76641558524888361269449095587, 5.47065187171852265626949231215, 5.95914563465824429620632747809, 6.70054889791097460341968358835, 7.950049633522063532948443519160
