L(s) = 1 | + 1.41·2-s + 1.00·4-s + i·5-s + 1.41i·7-s − 9-s + 1.41i·10-s − i·11-s + 1.41·13-s + 2.00i·14-s − 0.999·16-s + (0.707 − 0.707i)17-s − 1.41·18-s + 1.00i·20-s − 1.41i·22-s − 25-s + 2.00·26-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.00·4-s + i·5-s + 1.41i·7-s − 9-s + 1.41i·10-s − i·11-s + 1.41·13-s + 2.00i·14-s − 0.999·16-s + (0.707 − 0.707i)17-s − 1.41·18-s + 1.00i·20-s − 1.41i·22-s − 25-s + 2.00·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.898677522\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.898677522\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 - 1.41T + T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 7 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.87093465225964614282417696289, −9.408568757578713143824533303501, −8.721653001991800492311717712479, −7.78044279630834357161746005069, −6.32512594752181163238591754801, −5.91182732057657695355072990235, −5.45349867564459517890285824469, −3.93203903799502463312140495610, −3.05729241707130334440321461941, −2.51832731590390115501648903528,
1.45005472088593294785645829334, 3.23507069891033790149478572720, 4.01830476440969357056147757354, 4.73112711089147378791942331024, 5.64514788086739520499846624008, 6.41232611826163809997606784041, 7.49070526903456678453802319108, 8.454538037164120430675107926840, 9.246595418042383397188020171306, 10.44786024917173909604862724037