L(s) = 1 | + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (1.09 − 1.25i)5-s + (−0.382 − 0.923i)8-s + i·9-s + (0.108 − 1.65i)10-s + (−1.65 + 1.10i)13-s + (−0.866 − 0.499i)16-s + (−0.630 + 1.85i)17-s + (0.608 + 0.793i)18-s + (−0.923 − 1.38i)20-s + (−0.230 − 1.75i)25-s + (−0.641 + 1.88i)26-s + (0.0726 − 0.108i)29-s + (−0.991 + 0.130i)32-s + ⋯ |
L(s) = 1 | + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (1.09 − 1.25i)5-s + (−0.382 − 0.923i)8-s + i·9-s + (0.108 − 1.65i)10-s + (−1.65 + 1.10i)13-s + (−0.866 − 0.499i)16-s + (−0.630 + 1.85i)17-s + (0.608 + 0.793i)18-s + (−0.923 − 1.38i)20-s + (−0.230 − 1.75i)25-s + (−0.641 + 1.88i)26-s + (0.0726 − 0.108i)29-s + (−0.991 + 0.130i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.593504751\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.593504751\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.793 + 0.608i)T \) |
| 193 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (-1.09 + 1.25i)T + (-0.130 - 0.991i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (1.65 - 1.10i)T + (0.382 - 0.923i)T^{2} \) |
| 17 | \( 1 + (0.630 - 1.85i)T + (-0.793 - 0.608i)T^{2} \) |
| 19 | \( 1 + (0.130 + 0.991i)T^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.0726 + 0.108i)T + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.258 + 0.965i)T^{2} \) |
| 37 | \( 1 + (-0.665 + 1.34i)T + (-0.608 - 0.793i)T^{2} \) |
| 41 | \( 1 + (-1.69 - 0.576i)T + (0.793 + 0.608i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 53 | \( 1 + (1.69 + 0.837i)T + (0.608 + 0.793i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.583 + 0.665i)T + (-0.130 + 0.991i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (0.284 + 0.576i)T + (-0.608 + 0.793i)T^{2} \) |
| 79 | \( 1 + (-0.130 + 0.991i)T^{2} \) |
| 83 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 89 | \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \) |
| 97 | \( 1 + (-1.53 - 1.17i)T + (0.258 + 0.965i)T^{2} \) |
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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
−10.35988008475861393692140986167, −9.585158963662942052419639848669, −8.984581166205286639715306222716, −7.78147166818829515352485031956, −6.47817104017422390799876499399, −5.65150804036874722717112497249, −4.79233894184359297722234156361, −4.24681793743350476998382924556, −2.30372572472568799329027068713, −1.77458625032616306547956977416,
2.69835023649434470594164621772, 2.90060114328725907837314534759, 4.55230118676507945507072376667, 5.52643131041539395159799079946, 6.32132991212048498568004614349, 7.08999391787933954292588984834, 7.65399826092376402001980978009, 9.214449413593543848326105520610, 9.734468538704966937908439017936, 10.74752353809727545996135505119