L(s) = 1 | + (−1 + i)5-s − 4i·7-s + (4 − 4i)11-s + (−3 − 3i)13-s + 6·17-s + (4 + 4i)19-s + 8i·23-s + 3i·25-s + (−3 − 3i)29-s + 4·31-s + (4 + 4i)35-s + (−1 + i)37-s − 2i·41-s + (4 − 4i)43-s + 8·47-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.447i)5-s − 1.51i·7-s + (1.20 − 1.20i)11-s + (−0.832 − 0.832i)13-s + 1.45·17-s + (0.917 + 0.917i)19-s + 1.66i·23-s + 0.600i·25-s + (−0.557 − 0.557i)29-s + 0.718·31-s + (0.676 + 0.676i)35-s + (−0.164 + 0.164i)37-s − 0.312i·41-s + (0.609 − 0.609i)43-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.923775256\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.923775256\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 - i)T - 5iT^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + (-4 + 4i)T - 11iT^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (-4 - 4i)T + 19iT^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + (3 + 3i)T + 29iT^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 + (-4 + 4i)T - 43iT^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (-7 + 7i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (-3 - 3i)T + 61iT^{2} \) |
| 67 | \( 1 + (8 + 8i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + (4 + 4i)T + 83iT^{2} \) |
| 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
−7.83325522394818634085343838664, −7.51429224173958393752440525860, −7.02694324900049830732507678505, −5.82878137476376961810194530978, −5.47134438700881188236844577146, −4.13317443937412625674993055332, −3.52885920743695155344578371168, −3.17909892509341223686192717639, −1.40604506014445785688187876719, −0.67235618725128835732445613211,
1.05434687190868088777362237911, 2.21054636841203996196304650140, 2.87940617735897175679002636910, 4.16867341776349188836633158924, 4.68331474333806996471422034994, 5.44025664860789588903894374593, 6.29741948909370274433204702134, 7.05469919678144182569605052107, 7.67837427351456774870628049411, 8.664777297859929740659437856465