L(s) = 1 | + (0.324 − 1.63i)3-s + (−1.63 − 0.675i)9-s + (1.92 − 0.382i)11-s + (−0.923 + 0.382i)17-s + (−0.541 − 1.30i)19-s + (−0.382 + 0.923i)25-s + (−0.707 + 1.05i)27-s − 3.26i·33-s + (0.216 − 0.324i)41-s + (−0.707 + 1.70i)43-s + (0.382 + 0.923i)49-s + (0.324 + 1.63i)51-s + (−2.30 + 0.458i)57-s + (−1.70 − 0.707i)59-s + 0.765·67-s + ⋯ |
L(s) = 1 | + (0.324 − 1.63i)3-s + (−1.63 − 0.675i)9-s + (1.92 − 0.382i)11-s + (−0.923 + 0.382i)17-s + (−0.541 − 1.30i)19-s + (−0.382 + 0.923i)25-s + (−0.707 + 1.05i)27-s − 3.26i·33-s + (0.216 − 0.324i)41-s + (−0.707 + 1.70i)43-s + (0.382 + 0.923i)49-s + (0.324 + 1.63i)51-s + (−2.30 + 0.458i)57-s + (−1.70 − 0.707i)59-s + 0.765·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.173147132\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.173147132\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (0.923 - 0.382i)T \) |
good | 3 | \( 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2} \) |
| 5 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 7 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (-1.92 + 0.382i)T + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 29 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.216 + 0.324i)T + (-0.382 - 0.923i)T^{2} \) |
| 43 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 - 0.765T + T^{2} \) |
| 71 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 97 | \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \) |
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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
−9.430674926503093629457355416268, −8.928104871991970811634606990536, −8.161877494327783058340799197353, −7.19007260383785694595394255419, −6.56516291930919857878906405658, −6.07393684233369239700547571941, −4.54125142504142117715840547795, −3.37427629176284116471336989063, −2.16281856214574428888954764660, −1.17402827776802813127080373166,
2.04352750444639202145941728991, 3.54933951838268856491929288069, 4.08467860956650495343330929083, 4.82401072366003051773939726583, 6.01423869519503728391485251935, 6.81996355089682467493508656336, 8.165579699356822145522386675483, 8.947655961596254733375079719556, 9.441584552440296966513504679363, 10.19901062578974031207765882897