| L(s) = 1 | − 0.142·2-s − 3-s − 1.97·4-s + 0.142·6-s + 1.55·7-s + 0.566·8-s + 9-s + 1.06·11-s + 1.97·12-s + 6.73·13-s − 0.221·14-s + 3.87·16-s − 4.71·17-s − 0.142·18-s + 3.03·19-s − 1.55·21-s − 0.151·22-s − 5.80·23-s − 0.566·24-s − 0.958·26-s − 27-s − 3.08·28-s + 4.16·29-s − 8.70·31-s − 1.68·32-s − 1.06·33-s + 0.670·34-s + ⋯ |
| L(s) = 1 | − 0.100·2-s − 0.577·3-s − 0.989·4-s + 0.0580·6-s + 0.588·7-s + 0.200·8-s + 0.333·9-s + 0.320·11-s + 0.571·12-s + 1.86·13-s − 0.0591·14-s + 0.969·16-s − 1.14·17-s − 0.0335·18-s + 0.696·19-s − 0.339·21-s − 0.0322·22-s − 1.21·23-s − 0.115·24-s − 0.188·26-s − 0.192·27-s − 0.582·28-s + 0.772·29-s − 1.56·31-s − 0.297·32-s − 0.184·33-s + 0.114·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 + 0.142T + 2T^{2} \) |
| 7 | \( 1 - 1.55T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 13 | \( 1 - 6.73T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 3.03T + 19T^{2} \) |
| 23 | \( 1 + 5.80T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 + 6.57T + 37T^{2} \) |
| 41 | \( 1 - 7.77T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 + 3.58T + 59T^{2} \) |
| 61 | \( 1 - 0.168T + 61T^{2} \) |
| 67 | \( 1 + 0.691T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 0.977T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−7.64906626385582817640190595624, −6.57875072506353712465040912817, −6.15364232829412725951929232563, −5.25972662913208637450328516725, −4.78524310244922531971164661129, −3.86514301861944596836919670306, −3.51213719202027200743194696843, −1.88581843968702646104103063022, −1.17987168640998715641910649125, 0,
1.17987168640998715641910649125, 1.88581843968702646104103063022, 3.51213719202027200743194696843, 3.86514301861944596836919670306, 4.78524310244922531971164661129, 5.25972662913208637450328516725, 6.15364232829412725951929232563, 6.57875072506353712465040912817, 7.64906626385582817640190595624
