L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + 1.00i·6-s − 7-s + (0.707 + 0.707i)8-s − 1.00i·9-s − 1.00·10-s + 1.41·11-s + (−0.707 − 0.707i)12-s + (0.707 − 0.707i)14-s + 1.00·15-s − 1.00·16-s + (0.707 + 0.707i)18-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + 1.00i·6-s − 7-s + (0.707 + 0.707i)8-s − 1.00i·9-s − 1.00·10-s + 1.41·11-s + (−0.707 − 0.707i)12-s + (0.707 − 0.707i)14-s + 1.00·15-s − 1.00·16-s + (0.707 + 0.707i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9424799919\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9424799919\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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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
−10.02602407564292278200628695749, −9.346071787385526531448975983188, −8.965905496260437022994651597748, −7.75825478239488436021663627963, −6.89878225182377923035173813460, −6.46664504832763742085721758931, −5.72969406582502122066008486287, −3.91169756674228815526789900147, −2.69753690623999916111458804712, −1.46609455537118567618431703134,
1.55736254204345694322269012688, 2.82393587298750721365292387016, 3.78666591594725017670674313178, 4.65672707454783916310996498452, 6.09861588897084043394820675623, 7.10712888537064358741699816559, 8.351172049655512445844971551239, 8.971656165972742709272504859571, 9.538774521510132410758318226946, 10.03653258249946448263029217227