L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.541 − 1.30i)5-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.541 + 1.30i)10-s − 1.00·16-s − 1.00·18-s + (1.30 − 0.541i)20-s + (−0.707 + 0.707i)25-s + (−0.541 − 1.30i)29-s + (0.707 + 0.707i)32-s + (0.707 + 0.707i)36-s + (−1.30 + 0.541i)37-s + (−1.30 − 0.541i)40-s + (0.541 − 1.30i)41-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.541 − 1.30i)5-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.541 + 1.30i)10-s − 1.00·16-s − 1.00·18-s + (1.30 − 0.541i)20-s + (−0.707 + 0.707i)25-s + (−0.541 − 1.30i)29-s + (0.707 + 0.707i)32-s + (0.707 + 0.707i)36-s + (−1.30 + 0.541i)37-s + (−1.30 − 0.541i)40-s + (0.541 − 1.30i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6298527204\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6298527204\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
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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
−9.604641777032126863168522650357, −8.960521620835851013857347758429, −8.269857636964103829336318063498, −7.51719854346146224919590754450, −6.57536472882265877275954721629, −5.16832130570221569520932717966, −4.20793446163963183863860167720, −3.55934054757446103978096363008, −1.96221347260148237199134226628, −0.74323389126179422601511388332,
1.76431244579556931700635892351, 3.08203415893466478513500146656, 4.35029249334929860157248360240, 5.38084626813150845815464027475, 6.45013250596938970202618493246, 7.17327618002241218645316118322, 7.60360792509367313076915986091, 8.508163179093383524084798205624, 9.497638579998251220139941357241, 10.33247392518356803211830796657