| L(s) = 1 | − 2.67·2-s + 0.481·3-s + 5.15·4-s − 1.28·6-s − 0.806·7-s − 8.44·8-s − 2.76·9-s − 3.67·11-s + 2.48·12-s + 13-s + 2.15·14-s + 12.2·16-s + 1.35·17-s + 7.40·18-s − 1.67·19-s − 0.387·21-s + 9.83·22-s − 6.48·23-s − 4.06·24-s − 2.67·26-s − 2.77·27-s − 4.15·28-s + 2.41·29-s − 5.28·31-s − 15.9·32-s − 1.76·33-s − 3.61·34-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.277·3-s + 2.57·4-s − 0.525·6-s − 0.304·7-s − 2.98·8-s − 0.922·9-s − 1.10·11-s + 0.716·12-s + 0.277·13-s + 0.576·14-s + 3.06·16-s + 0.327·17-s + 1.74·18-s − 0.384·19-s − 0.0846·21-s + 2.09·22-s − 1.35·23-s − 0.829·24-s − 0.524·26-s − 0.534·27-s − 0.785·28-s + 0.449·29-s − 0.949·31-s − 2.81·32-s − 0.307·33-s − 0.619·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 - 0.481T + 3T^{2} \) |
| 7 | \( 1 + 0.806T + 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 + 1.67T + 19T^{2} \) |
| 23 | \( 1 + 6.48T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 + 8.31T + 41T^{2} \) |
| 43 | \( 1 - 6.79T + 43T^{2} \) |
| 47 | \( 1 + 3.19T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 - 5.98T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + 9.89T + 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 2.26T + 79T^{2} \) |
| 83 | \( 1 + 3.84T + 83T^{2} \) |
| 89 | \( 1 - 2.77T + 89T^{2} \) |
| 97 | \( 1 - 1.87T + 97T^{2} \) |
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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
−10.79267788413257738165525226841, −10.10395974066581299622102063288, −9.213066681396800046603637039199, −8.299837515005395000318143135891, −7.82297581536127608376053453537, −6.61050142773069633317339048290, −5.60338641767849978921876689641, −3.22498838443237238631056320219, −2.07167395330054850356517475875, 0,
2.07167395330054850356517475875, 3.22498838443237238631056320219, 5.60338641767849978921876689641, 6.61050142773069633317339048290, 7.82297581536127608376053453537, 8.299837515005395000318143135891, 9.213066681396800046603637039199, 10.10395974066581299622102063288, 10.79267788413257738165525226841
