L(s) = 1 | + 2-s − 0.687·3-s + 4-s − 5-s − 0.687·6-s + 3.00·7-s + 8-s − 2.52·9-s − 10-s + 3.24·11-s − 0.687·12-s + 4.42·13-s + 3.00·14-s + 0.687·15-s + 16-s + 7.05·17-s − 2.52·18-s + 3.61·19-s − 20-s − 2.06·21-s + 3.24·22-s − 0.687·24-s + 25-s + 4.42·26-s + 3.80·27-s + 3.00·28-s − 3.99·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.397·3-s + 0.5·4-s − 0.447·5-s − 0.280·6-s + 1.13·7-s + 0.353·8-s − 0.842·9-s − 0.316·10-s + 0.977·11-s − 0.198·12-s + 1.22·13-s + 0.803·14-s + 0.177·15-s + 0.250·16-s + 1.71·17-s − 0.595·18-s + 0.829·19-s − 0.223·20-s − 0.451·21-s + 0.690·22-s − 0.140·24-s + 0.200·25-s + 0.867·26-s + 0.731·27-s + 0.568·28-s − 0.741·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.378638690\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.378638690\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + 0.687T + 3T^{2} \) |
| 7 | \( 1 - 3.00T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 - 4.42T + 13T^{2} \) |
| 17 | \( 1 - 7.05T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 - 1.99T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 6.41T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 7.64T + 59T^{2} \) |
| 61 | \( 1 + 7.00T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 - 8.14T + 71T^{2} \) |
| 73 | \( 1 - 6.95T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 - 4.03T + 97T^{2} \) |
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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
−8.046355523617301329825814388438, −7.54476517426705068577705033773, −6.55824722914434212019237741963, −5.89687362972565022214274882071, −5.27794591659126184130884778460, −4.64899703952018018560250040397, −3.54363397697979269798590452061, −3.29833000724994474957114987979, −1.75445507119023894415343589911, −1.00213288693484820238404037074,
1.00213288693484820238404037074, 1.75445507119023894415343589911, 3.29833000724994474957114987979, 3.54363397697979269798590452061, 4.64899703952018018560250040397, 5.27794591659126184130884778460, 5.89687362972565022214274882071, 6.55824722914434212019237741963, 7.54476517426705068577705033773, 8.046355523617301329825814388438