| L(s) = 1 | − 2.09·2-s − 0.807·3-s + 2.36·4-s + 1.68·6-s − 0.424·7-s − 0.773·8-s − 2.34·9-s + 5.89·11-s − 1.91·12-s − 4.73·13-s + 0.887·14-s − 3.12·16-s + 3.30·17-s + 4.90·18-s + 4.30·19-s + 0.342·21-s − 12.3·22-s − 6.48·23-s + 0.624·24-s + 9.90·26-s + 4.31·27-s − 1.00·28-s − 5.81·29-s + 10.3·31-s + 8.07·32-s − 4.76·33-s − 6.90·34-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 0.466·3-s + 1.18·4-s + 0.689·6-s − 0.160·7-s − 0.273·8-s − 0.782·9-s + 1.77·11-s − 0.552·12-s − 1.31·13-s + 0.237·14-s − 0.780·16-s + 0.800·17-s + 1.15·18-s + 0.987·19-s + 0.0747·21-s − 2.62·22-s − 1.35·23-s + 0.127·24-s + 1.94·26-s + 0.831·27-s − 0.190·28-s − 1.07·29-s + 1.85·31-s + 1.42·32-s − 0.829·33-s − 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5448522548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5448522548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 3 | \( 1 + 0.807T + 3T^{2} \) |
| 7 | \( 1 + 0.424T + 7T^{2} \) |
| 11 | \( 1 - 5.89T + 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 + 6.48T + 23T^{2} \) |
| 29 | \( 1 + 5.81T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 6.61T + 43T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 1.21T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 + 5.54T + 83T^{2} \) |
| 89 | \( 1 + 2.77T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−9.897207487609831939155722737153, −9.026903337554772771390554369313, −8.292125342100376084420838500151, −7.42990396674836536907734794652, −6.68591312079824386016861322525, −5.81712222040443926998737616577, −4.69227334535340991081076488485, −3.38294260956456009245292468019, −1.97876128602272218816172441406, −0.70845664832646883266429238246,
0.70845664832646883266429238246, 1.97876128602272218816172441406, 3.38294260956456009245292468019, 4.69227334535340991081076488485, 5.81712222040443926998737616577, 6.68591312079824386016861322525, 7.42990396674836536907734794652, 8.292125342100376084420838500151, 9.026903337554772771390554369313, 9.897207487609831939155722737153
