| L(s) = 1 | + 2-s − i·3-s + 4-s − i·6-s + 3·7-s + 8-s − 9-s + 5i·11-s − i·12-s + (2 + 3i)13-s + 3·14-s + 16-s − 7i·17-s − 18-s + 4i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 1.13·7-s + 0.353·8-s − 0.333·9-s + 1.50i·11-s − 0.288i·12-s + (0.554 + 0.832i)13-s + 0.801·14-s + 0.250·16-s − 1.69i·17-s − 0.235·18-s + 0.917i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.204240674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.204240674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2 - 3i)T \) |
| good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 5iT - 11T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + 11iT - 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.174083056813021041848900294074, −8.221386122725656731195566504438, −7.33189067866850681642682591467, −7.05557250673826687196833987821, −5.95174744379117662786101475525, −4.98319389822163979942693905356, −4.53750356151393784038598312343, −3.35178702918451943111584424923, −2.09336847200769477098002337447, −1.45491289552526550486262302402,
1.02595622375540356416283476511, 2.49153215459591138965778677445, 3.44662618665729491112720039250, 4.27924155719094403777257246368, 5.05038657882760264012725781375, 5.91875119374335620317214570393, 6.41317365972876743535271740190, 7.84962674250234991788385617270, 8.362753038381910379653896399934, 8.900708294718218593908172421737
