| L(s) = 1 | − 0.347i·3-s − 3.41i·7-s + 2.87·9-s + 2.41·11-s − 6.29i·13-s − 2.34i·17-s − 19-s − 1.18·21-s − 2.49i·23-s − 2.04i·27-s + 8.17·29-s − 2.77·31-s − 0.837i·33-s − 0.977i·37-s − 2.18·39-s + ⋯ |
| L(s) = 1 | − 0.200i·3-s − 1.28i·7-s + 0.959·9-s + 0.727·11-s − 1.74i·13-s − 0.569i·17-s − 0.229·19-s − 0.258·21-s − 0.519i·23-s − 0.392i·27-s + 1.51·29-s − 0.498·31-s − 0.145i·33-s − 0.160i·37-s − 0.349·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.989329724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.989329724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.347iT - 3T^{2} \) |
| 7 | \( 1 + 3.41iT - 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 6.29iT - 13T^{2} \) |
| 17 | \( 1 + 2.34iT - 17T^{2} \) |
| 23 | \( 1 + 2.49iT - 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 0.977iT - 37T^{2} \) |
| 41 | \( 1 + 3.49T + 41T^{2} \) |
| 43 | \( 1 - 2.75iT - 43T^{2} \) |
| 47 | \( 1 - 6.29iT - 47T^{2} \) |
| 53 | \( 1 + 2.38iT - 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 2.41iT - 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 + 1.81iT - 73T^{2} \) |
| 79 | \( 1 - 5.04T + 79T^{2} \) |
| 83 | \( 1 - 8.07iT - 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 3.09iT - 97T^{2} \) |
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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
−7.975727670159443995970786172481, −7.62005158822670028803309901470, −6.79034462353144567564949155096, −6.27863755806285691844212619052, −5.10173832176468083521331896749, −4.42115863555398423969183010761, −3.65071992631756074727208803121, −2.76552139917970842833926793925, −1.35528109140582765790626247289, −0.62440370078852320082160612569,
1.50601740205259182684811960351, 2.12171625357295156442403259898, 3.36571280805303564520261683484, 4.24516276357880618264776846238, 4.82532741913189598420109175174, 5.85505911803266729765354787691, 6.55993858303700047524542450020, 7.07356802786268792572702282622, 8.154230513918400660110746195226, 8.942763432709325135908630866790
