| L(s) = 1 | + 1.01·3-s + 0.217·5-s + 2.74·7-s − 1.97·9-s − 1.10·11-s + 3.99·13-s + 0.219·15-s − 2.81·17-s − 8.43·19-s + 2.78·21-s − 1.44·23-s − 4.95·25-s − 5.03·27-s − 5.00·29-s − 3.45·31-s − 1.11·33-s + 0.597·35-s + 2.96·37-s + 4.04·39-s − 11.5·41-s − 5.07·43-s − 0.429·45-s + 8.95·47-s + 0.555·49-s − 2.84·51-s − 2.67·53-s − 0.240·55-s + ⋯ |
| L(s) = 1 | + 0.584·3-s + 0.0971·5-s + 1.03·7-s − 0.658·9-s − 0.333·11-s + 1.10·13-s + 0.0567·15-s − 0.683·17-s − 1.93·19-s + 0.606·21-s − 0.301·23-s − 0.990·25-s − 0.968·27-s − 0.928·29-s − 0.620·31-s − 0.194·33-s + 0.100·35-s + 0.487·37-s + 0.647·39-s − 1.80·41-s − 0.774·43-s − 0.0640·45-s + 1.30·47-s + 0.0793·49-s − 0.399·51-s − 0.367·53-s − 0.0324·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 - 0.217T + 5T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 + 8.43T + 19T^{2} \) |
| 23 | \( 1 + 1.44T + 23T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 - 2.96T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 5.07T + 43T^{2} \) |
| 47 | \( 1 - 8.95T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 - 3.44T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4.43T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 5.55T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 2.92T + 97T^{2} \) |
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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
−8.155221864994629413201702068004, −7.54811321498126270013033008106, −6.43445697474110199744977298543, −5.88378507623541834164081025135, −4.99738876303499431009940249758, −4.12682303081906551045817828786, −3.45141509478160475883082190090, −2.21614335610604889972346402159, −1.77499255526640895610597875253, 0,
1.77499255526640895610597875253, 2.21614335610604889972346402159, 3.45141509478160475883082190090, 4.12682303081906551045817828786, 4.99738876303499431009940249758, 5.88378507623541834164081025135, 6.43445697474110199744977298543, 7.54811321498126270013033008106, 8.155221864994629413201702068004
