| L(s) = 1 | + 2.59·2-s + 1.20·3-s + 4.74·4-s + 3.13·6-s + 0.744·7-s + 7.12·8-s − 1.54·9-s + 6.28·11-s + 5.71·12-s + 4.06·13-s + 1.93·14-s + 9.02·16-s + 1.60·17-s − 4.01·18-s + 2.14·19-s + 0.897·21-s + 16.3·22-s − 9.25·23-s + 8.59·24-s + 10.5·26-s − 5.48·27-s + 3.53·28-s + 3.13·29-s − 3.15·31-s + 9.17·32-s + 7.57·33-s + 4.15·34-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.695·3-s + 2.37·4-s + 1.27·6-s + 0.281·7-s + 2.52·8-s − 0.515·9-s + 1.89·11-s + 1.65·12-s + 1.12·13-s + 0.516·14-s + 2.25·16-s + 0.388·17-s − 0.947·18-s + 0.491·19-s + 0.195·21-s + 3.48·22-s − 1.92·23-s + 1.75·24-s + 2.07·26-s − 1.05·27-s + 0.667·28-s + 0.581·29-s − 0.566·31-s + 1.62·32-s + 1.31·33-s + 0.712·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.797679731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.797679731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 3 | \( 1 - 1.20T + 3T^{2} \) |
| 7 | \( 1 - 0.744T + 7T^{2} \) |
| 11 | \( 1 - 6.28T + 11T^{2} \) |
| 13 | \( 1 - 4.06T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 23 | \( 1 + 9.25T + 23T^{2} \) |
| 29 | \( 1 - 3.13T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 + 4.19T + 37T^{2} \) |
| 41 | \( 1 + 4.52T + 41T^{2} \) |
| 43 | \( 1 + 6.99T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 3.71T + 53T^{2} \) |
| 59 | \( 1 - 3.34T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 1.64T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.74T + 83T^{2} \) |
| 89 | \( 1 + 7.95T + 89T^{2} \) |
| 97 | \( 1 - 5.49T + 97T^{2} \) |
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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
−8.053154266233791052086863439525, −7.04175548207397866359502796380, −6.41672858328065585629105686586, −5.89381214428857316065976240264, −5.18477606063700190823359333910, −4.15550603325916963238695765866, −3.68795883511494264094841482174, −3.25216199391103844217759163150, −2.09547666587401544829821476218, −1.43589432747494202326951144168,
1.43589432747494202326951144168, 2.09547666587401544829821476218, 3.25216199391103844217759163150, 3.68795883511494264094841482174, 4.15550603325916963238695765866, 5.18477606063700190823359333910, 5.89381214428857316065976240264, 6.41672858328065585629105686586, 7.04175548207397866359502796380, 8.053154266233791052086863439525
