L(s) = 1 | + 2-s + 3.44·3-s + 4-s + 2.54·5-s + 3.44·6-s + 0.833·7-s + 8-s + 8.84·9-s + 2.54·10-s + 11-s + 3.44·12-s − 5.94·13-s + 0.833·14-s + 8.74·15-s + 16-s − 2.38·17-s + 8.84·18-s − 4.98·19-s + 2.54·20-s + 2.86·21-s + 22-s + 5.23·23-s + 3.44·24-s + 1.45·25-s − 5.94·26-s + 20.1·27-s + 0.833·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.98·3-s + 0.5·4-s + 1.13·5-s + 1.40·6-s + 0.314·7-s + 0.353·8-s + 2.94·9-s + 0.803·10-s + 0.301·11-s + 0.993·12-s − 1.64·13-s + 0.222·14-s + 2.25·15-s + 0.250·16-s − 0.579·17-s + 2.08·18-s − 1.14·19-s + 0.567·20-s + 0.625·21-s + 0.213·22-s + 1.09·23-s + 0.702·24-s + 0.290·25-s − 1.16·26-s + 3.87·27-s + 0.157·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.892805110\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.892805110\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
good | 3 | \( 1 - 3.44T + 3T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 7 | \( 1 - 0.833T + 7T^{2} \) |
| 13 | \( 1 + 5.94T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + 1.29T + 29T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 - 0.0979T + 37T^{2} \) |
| 41 | \( 1 - 0.398T + 41T^{2} \) |
| 43 | \( 1 + 5.83T + 43T^{2} \) |
| 47 | \( 1 + 1.15T + 47T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 + 3.71T + 71T^{2} \) |
| 73 | \( 1 + 9.91T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + 1.51T + 83T^{2} \) |
| 89 | \( 1 - 7.21T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
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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
−8.421544331762194607319959666631, −7.62300885327330268213196573509, −6.97382219818391106387406122467, −6.37875708494613171644720735025, −5.06535142351175458757957406637, −4.60470802765105265958603264804, −3.68141344312283664818522332472, −2.77136406394175154064281434810, −2.19266573502009907448080260455, −1.63381201387813156701257282554,
1.63381201387813156701257282554, 2.19266573502009907448080260455, 2.77136406394175154064281434810, 3.68141344312283664818522332472, 4.60470802765105265958603264804, 5.06535142351175458757957406637, 6.37875708494613171644720735025, 6.97382219818391106387406122467, 7.62300885327330268213196573509, 8.421544331762194607319959666631