L(s) = 1 | + (0.195 − 0.980i)2-s + (−0.831 + 0.555i)3-s + (−0.923 − 0.382i)4-s + (−0.980 + 0.195i)5-s + (0.382 + 0.923i)6-s + (−0.555 + 0.831i)8-s + (0.382 − 0.923i)9-s + i·10-s + (0.980 − 0.195i)12-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)16-s + (1.38 + 1.38i)17-s + (−0.831 − 0.555i)18-s + (−0.216 + 1.08i)19-s + (0.980 + 0.195i)20-s + ⋯ |
L(s) = 1 | + (0.195 − 0.980i)2-s + (−0.831 + 0.555i)3-s + (−0.923 − 0.382i)4-s + (−0.980 + 0.195i)5-s + (0.382 + 0.923i)6-s + (−0.555 + 0.831i)8-s + (0.382 − 0.923i)9-s + i·10-s + (0.980 − 0.195i)12-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)16-s + (1.38 + 1.38i)17-s + (−0.831 − 0.555i)18-s + (−0.216 + 1.08i)19-s + (0.980 + 0.195i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5761187789\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5761187789\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.195 + 0.980i)T \) |
| 3 | \( 1 + (0.831 - 0.555i)T \) |
| 5 | \( 1 + (0.980 - 0.195i)T \) |
good | 7 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 13 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 17 | \( 1 + (-1.38 - 1.38i)T + iT^{2} \) |
| 19 | \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \) |
| 23 | \( 1 + (-0.360 - 0.149i)T + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 - 1.84iT - T^{2} \) |
| 37 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 47 | \( 1 + (1.17 + 1.17i)T + iT^{2} \) |
| 53 | \( 1 + (-0.425 + 0.636i)T + (-0.382 - 0.923i)T^{2} \) |
| 59 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 61 | \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (1 - i)T - iT^{2} \) |
| 83 | \( 1 + (0.149 - 0.750i)T + (-0.923 - 0.382i)T^{2} \) |
| 89 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 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
−10.33774114719560779646766128631, −9.942783560543665735967359922196, −8.689667348053783520899213244762, −8.009478589266112202510599108499, −6.69689458136161567201320183952, −5.66427406032825240641464185596, −4.86131603878145511114976204260, −3.78044816078384278119308604387, −3.36381605688809156043586814461, −1.36720292957417180416800769734,
0.67176482383833027966779227335, 3.02550938586417821411225580870, 4.37768604550014607337306900475, 5.03828008990653321950869295280, 5.91935822623016012779784198958, 6.95697665106169369471655950561, 7.48581755536491249874609974846, 8.136507024723783556098791663471, 9.183808458331550396631175687170, 10.08689084967034117086627930423