| L(s) = 1 | + 2.28·2-s + 3-s + 3.24·4-s − 3.89·5-s + 2.28·6-s − 1.35·7-s + 2.84·8-s + 9-s − 8.91·10-s + 1.39·11-s + 3.24·12-s − 13-s − 3.09·14-s − 3.89·15-s + 0.0256·16-s − 0.168·17-s + 2.28·18-s + 3.52·19-s − 12.6·20-s − 1.35·21-s + 3.18·22-s − 4.03·23-s + 2.84·24-s + 10.1·25-s − 2.28·26-s + 27-s − 4.38·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 0.577·3-s + 1.62·4-s − 1.74·5-s + 0.934·6-s − 0.511·7-s + 1.00·8-s + 0.333·9-s − 2.82·10-s + 0.419·11-s + 0.935·12-s − 0.277·13-s − 0.828·14-s − 1.00·15-s + 0.00641·16-s − 0.0407·17-s + 0.539·18-s + 0.808·19-s − 2.82·20-s − 0.295·21-s + 0.678·22-s − 0.840·23-s + 0.580·24-s + 2.03·25-s − 0.449·26-s + 0.192·27-s − 0.829·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 5 | \( 1 + 3.89T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 17 | \( 1 + 0.168T + 17T^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 - 2.45T + 41T^{2} \) |
| 43 | \( 1 + 9.77T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 6.21T + 53T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 + 8.58T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 + 6.33T + 73T^{2} \) |
| 79 | \( 1 + 0.208T + 79T^{2} \) |
| 83 | \( 1 - 3.54T + 83T^{2} \) |
| 89 | \( 1 - 8.49T + 89T^{2} \) |
| 97 | \( 1 - 2.54T + 97T^{2} \) |
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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
−7.73438980299107025802195221041, −7.30623564938957263243916575208, −6.61999899589953058662931312678, −5.67971244251724817694497496846, −4.79236703590094988410695626124, −4.12682215356161207581707553768, −3.43805635835703019068403013889, −3.18508982910903755280157756911, −1.86091020804325146333062815442, 0,
1.86091020804325146333062815442, 3.18508982910903755280157756911, 3.43805635835703019068403013889, 4.12682215356161207581707553768, 4.79236703590094988410695626124, 5.67971244251724817694497496846, 6.61999899589953058662931312678, 7.30623564938957263243916575208, 7.73438980299107025802195221041
