| L(s) = 1 | − 1.75·3-s + 4-s − 1.35·5-s + 2.09·9-s − 0.803·11-s − 1.75·12-s − 1.97·13-s + 2.38·15-s + 16-s + 1.57·19-s − 1.35·20-s + 0.834·25-s − 1.92·27-s + 1.41·33-s + 2.09·36-s + 3.46·39-s + 0.490·41-s − 0.803·44-s − 2.83·45-s − 1.75·48-s + 49-s − 1.97·52-s + 1.89·53-s + 1.08·55-s − 2.77·57-s − 1.97·59-s + 2.38·60-s + ⋯ |
| L(s) = 1 | − 1.75·3-s + 4-s − 1.35·5-s + 2.09·9-s − 0.803·11-s − 1.75·12-s − 1.97·13-s + 2.38·15-s + 16-s + 1.57·19-s − 1.35·20-s + 0.834·25-s − 1.92·27-s + 1.41·33-s + 2.09·36-s + 3.46·39-s + 0.490·41-s − 0.803·44-s − 2.83·45-s − 1.75·48-s + 49-s − 1.97·52-s + 1.89·53-s + 1.08·55-s − 2.77·57-s − 1.97·59-s + 2.38·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5225267265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5225267265\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2339 | \( 1+O(T) \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 1.75T + T^{2} \) |
| 5 | \( 1 + 1.35T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.803T + T^{2} \) |
| 13 | \( 1 + 1.97T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.57T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.490T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.89T + T^{2} \) |
| 59 | \( 1 + 1.97T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.490T + T^{2} \) |
| 71 | \( 1 + 0.165T + T^{2} \) |
| 73 | \( 1 - 1.09T + T^{2} \) |
| 79 | \( 1 - 0.490T + T^{2} \) |
| 83 | \( 1 - 1.57T + T^{2} \) |
| 89 | \( 1 + 0.165T + T^{2} \) |
| 97 | \( 1 - 1.89T + T^{2} \) |
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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
−9.447218682042925082499075904266, −7.85102983021580003100222367122, −7.42888355791315700254114865455, −7.05738881136209306799325322526, −6.00904464418107634460572999839, −5.20684941341147690225819479492, −4.70908560354535666646043719143, −3.51686805570511521382048065779, −2.37345953529544673061611154845, −0.72349883976655783718396239623,
0.72349883976655783718396239623, 2.37345953529544673061611154845, 3.51686805570511521382048065779, 4.70908560354535666646043719143, 5.20684941341147690225819479492, 6.00904464418107634460572999839, 7.05738881136209306799325322526, 7.42888355791315700254114865455, 7.85102983021580003100222367122, 9.447218682042925082499075904266
