L(s) = 1 | + 3-s + 1.12·7-s + 9-s − 0.781·11-s + 6.90·13-s − 17-s − 4.99·19-s + 1.12·21-s + 2.69·23-s + 27-s − 4.48·29-s + 5.36·31-s − 0.781·33-s − 1.87·37-s + 6.90·39-s + 12.1·41-s − 4.10·43-s + 12.1·47-s − 5.73·49-s − 51-s + 7.12·53-s − 4.99·57-s + 8.61·59-s − 13.0·61-s + 1.12·63-s − 8.61·67-s + 2.69·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.424·7-s + 0.333·9-s − 0.235·11-s + 1.91·13-s − 0.242·17-s − 1.14·19-s + 0.245·21-s + 0.562·23-s + 0.192·27-s − 0.833·29-s + 0.963·31-s − 0.136·33-s − 0.308·37-s + 1.10·39-s + 1.90·41-s − 0.625·43-s + 1.76·47-s − 0.819·49-s − 0.140·51-s + 0.978·53-s − 0.660·57-s + 1.12·59-s − 1.66·61-s + 0.141·63-s − 1.05·67-s + 0.324·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.853658878\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.853658878\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 + 0.781T + 11T^{2} \) |
| 13 | \( 1 - 6.90T + 13T^{2} \) |
| 19 | \( 1 + 4.99T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 4.10T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 - 8.61T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 8.61T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−8.264630366534032086901416333290, −7.70944008971150924693770881347, −6.75137904908260658846669924449, −6.14262763647478122432066712986, −5.34346001837206476991025358882, −4.29255802487438778695944304928, −3.83090617083835083048326069635, −2.82432007890946143206135795104, −1.93284826448060110361544688751, −0.937155415598296276944763775701,
0.937155415598296276944763775701, 1.93284826448060110361544688751, 2.82432007890946143206135795104, 3.83090617083835083048326069635, 4.29255802487438778695944304928, 5.34346001837206476991025358882, 6.14262763647478122432066712986, 6.75137904908260658846669924449, 7.70944008971150924693770881347, 8.264630366534032086901416333290