L(s) = 1 | + 0.334·2-s + 0.0869·3-s − 1.88·4-s + 0.0290·6-s + 2.52·7-s − 1.30·8-s − 2.99·9-s + 2.78·11-s − 0.164·12-s + 6.58·13-s + 0.845·14-s + 3.34·16-s + 2.34·17-s − 1.00·18-s − 3.77·19-s + 0.219·21-s + 0.931·22-s + 2.43·23-s − 0.113·24-s + 2.20·26-s − 0.520·27-s − 4.77·28-s − 1.01·29-s − 2.06·31-s + 3.71·32-s + 0.242·33-s + 0.783·34-s + ⋯ |
L(s) = 1 | + 0.236·2-s + 0.0501·3-s − 0.944·4-s + 0.0118·6-s + 0.955·7-s − 0.459·8-s − 0.997·9-s + 0.840·11-s − 0.0473·12-s + 1.82·13-s + 0.226·14-s + 0.835·16-s + 0.568·17-s − 0.235·18-s − 0.865·19-s + 0.0479·21-s + 0.198·22-s + 0.508·23-s − 0.0230·24-s + 0.431·26-s − 0.100·27-s − 0.902·28-s − 0.187·29-s − 0.370·31-s + 0.657·32-s + 0.0421·33-s + 0.134·34-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
\(L(1)\) |
\(\approx\) |
\(2.133167407\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.133167407\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.334T + 2T^{2} \) |
| 3 | \( 1 - 0.0869T + 3T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 19 | \( 1 + 3.77T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 - 5.93T + 47T^{2} \) |
| 53 | \( 1 + 6.71T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 2.99T + 61T^{2} \) |
| 67 | \( 1 + 5.81T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.368319641773672578852778224965, −7.56729144656080922042952577946, −6.35237773895579399668517325390, −5.93035821268629824110171164847, −5.19684570542048144781117275214, −4.37039765650521712243236256434, −3.76641850874111926254300278278, −3.02633317395570777926674438419, −1.68508650858525894738566107274, −0.794735290914990998217949997788,
0.794735290914990998217949997788, 1.68508650858525894738566107274, 3.02633317395570777926674438419, 3.76641850874111926254300278278, 4.37039765650521712243236256434, 5.19684570542048144781117275214, 5.93035821268629824110171164847, 6.35237773895579399668517325390, 7.56729144656080922042952577946, 8.368319641773672578852778224965