| L(s) = 1 | − 3.28·3-s − 3.08·5-s + 0.442·7-s + 7.76·9-s + 1.35·11-s − 2.73·13-s + 10.1·15-s + 4.78·17-s + 2.61·19-s − 1.45·21-s + 3.60·23-s + 4.52·25-s − 15.6·27-s − 5.20·29-s − 0.413·31-s − 4.44·33-s − 1.36·35-s + 0.790·37-s + 8.98·39-s − 2.32·41-s + 4.33·43-s − 23.9·45-s − 1.17·47-s − 6.80·49-s − 15.7·51-s − 5.33·53-s − 4.18·55-s + ⋯ |
| L(s) = 1 | − 1.89·3-s − 1.37·5-s + 0.167·7-s + 2.58·9-s + 0.408·11-s − 0.759·13-s + 2.61·15-s + 1.16·17-s + 0.599·19-s − 0.316·21-s + 0.752·23-s + 0.904·25-s − 3.01·27-s − 0.967·29-s − 0.0743·31-s − 0.774·33-s − 0.230·35-s + 0.129·37-s + 1.43·39-s − 0.363·41-s + 0.661·43-s − 3.57·45-s − 0.170·47-s − 0.972·49-s − 2.19·51-s − 0.732·53-s − 0.564·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5448847619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5448847619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 3.28T + 3T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 7 | \( 1 - 0.442T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 + 0.413T + 31T^{2} \) |
| 37 | \( 1 - 0.790T + 37T^{2} \) |
| 41 | \( 1 + 2.32T + 41T^{2} \) |
| 43 | \( 1 - 4.33T + 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 + 5.33T + 53T^{2} \) |
| 59 | \( 1 - 7.03T + 59T^{2} \) |
| 61 | \( 1 + 6.81T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 + 4.57T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 7.04T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.152306245560984247209371351489, −7.40587806890378876870275855475, −7.12309396660044047680200072394, −6.16172563189444963082771489992, −5.38636798621105334544111994936, −4.80923740758778816774547918516, −4.08884200315539682770847815085, −3.25034175956724943839629693991, −1.47804287759632196065966307237, −0.50087198348316043327413989436,
0.50087198348316043327413989436, 1.47804287759632196065966307237, 3.25034175956724943839629693991, 4.08884200315539682770847815085, 4.80923740758778816774547918516, 5.38636798621105334544111994936, 6.16172563189444963082771489992, 7.12309396660044047680200072394, 7.40587806890378876870275855475, 8.152306245560984247209371351489
