| L(s) = 1 | − 2-s + 2.92·3-s + 4-s + 1.30·5-s − 2.92·6-s − 0.0744·7-s − 8-s + 5.56·9-s − 1.30·10-s − 5.65·11-s + 2.92·12-s + 0.0744·14-s + 3.83·15-s + 16-s − 4.00·17-s − 5.56·18-s − 2.23·19-s + 1.30·20-s − 0.217·21-s + 5.65·22-s + 23-s − 2.92·24-s − 3.28·25-s + 7.49·27-s − 0.0744·28-s + 1.67·29-s − 3.83·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.68·3-s + 0.5·4-s + 0.585·5-s − 1.19·6-s − 0.0281·7-s − 0.353·8-s + 1.85·9-s − 0.414·10-s − 1.70·11-s + 0.844·12-s + 0.0198·14-s + 0.989·15-s + 0.250·16-s − 0.972·17-s − 1.31·18-s − 0.512·19-s + 0.292·20-s − 0.0475·21-s + 1.20·22-s + 0.208·23-s − 0.597·24-s − 0.656·25-s + 1.44·27-s − 0.0140·28-s + 0.310·29-s − 0.699·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7774 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7774 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 + 0.0744T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 17 | \( 1 + 4.00T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 + 8.30T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 - 3.97T + 43T^{2} \) |
| 47 | \( 1 + 2.02T + 47T^{2} \) |
| 53 | \( 1 - 5.91T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 9.43T + 61T^{2} \) |
| 67 | \( 1 - 8.32T + 67T^{2} \) |
| 71 | \( 1 + 6.76T + 71T^{2} \) |
| 73 | \( 1 + 2.97T + 73T^{2} \) |
| 79 | \( 1 - 0.878T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 + 7.71T + 97T^{2} \) |
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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
−7.75925153396923305965936480321, −7.17973471113352318736526101407, −6.32780611278410047621352294490, −5.45519675682762924475433692873, −4.57130341538513705472443186841, −3.62707953870245869684240808567, −2.78337731706362846850480772991, −2.27862125835957135677677095536, −1.65655895987127195804766047055, 0,
1.65655895987127195804766047055, 2.27862125835957135677677095536, 2.78337731706362846850480772991, 3.62707953870245869684240808567, 4.57130341538513705472443186841, 5.45519675682762924475433692873, 6.32780611278410047621352294490, 7.17973471113352318736526101407, 7.75925153396923305965936480321
