| L(s) = 1 | − 2.80·2-s + 1.73·3-s + 5.84·4-s − 4.85·6-s − 7-s − 10.7·8-s − 0.000289·9-s + 1.96·11-s + 10.1·12-s − 3.06·13-s + 2.80·14-s + 18.5·16-s + 5.01·17-s + 0.000811·18-s − 0.164·19-s − 1.73·21-s − 5.51·22-s − 23-s − 18.6·24-s + 8.57·26-s − 5.19·27-s − 5.84·28-s − 1.60·29-s − 1.84·31-s − 30.3·32-s + 3.40·33-s − 14.0·34-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.999·3-s + 2.92·4-s − 1.98·6-s − 0.377·7-s − 3.81·8-s − 9.65e − 5·9-s + 0.593·11-s + 2.92·12-s − 0.849·13-s + 0.748·14-s + 4.62·16-s + 1.21·17-s + 0.000191·18-s − 0.0377·19-s − 0.377·21-s − 1.17·22-s − 0.208·23-s − 3.81·24-s + 1.68·26-s − 1.00·27-s − 1.10·28-s − 0.297·29-s − 0.330·31-s − 5.35·32-s + 0.593·33-s − 2.41·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 3 | \( 1 - 1.73T + 3T^{2} \) |
| 11 | \( 1 - 1.96T + 11T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 + 0.164T + 19T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 + 7.42T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 - 4.25T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 7.76T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 8.46T + 89T^{2} \) |
| 97 | \( 1 + 4.55T + 97T^{2} \) |
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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
−8.192564228426528985549565183257, −7.58123642699781995045900771079, −7.11541023921814389768638202150, −6.21835022148628638243555123629, −5.44824351143914118368991781925, −3.70055583846696881091034127704, −3.01624756009386743270083088654, −2.25029056730156071900050376216, −1.34713112479504693680172247709, 0,
1.34713112479504693680172247709, 2.25029056730156071900050376216, 3.01624756009386743270083088654, 3.70055583846696881091034127704, 5.44824351143914118368991781925, 6.21835022148628638243555123629, 7.11541023921814389768638202150, 7.58123642699781995045900771079, 8.192564228426528985549565183257
