L(s) = 1 | + (−1.33 + 0.456i)2-s + (−1.70 − 0.277i)3-s + (1.58 − 1.22i)4-s + (−0.459 − 0.459i)5-s + (2.41 − 0.410i)6-s + 7-s + (−1.55 + 2.35i)8-s + (2.84 + 0.947i)9-s + (0.824 + 0.405i)10-s + (−1.43 + 1.43i)11-s + (−3.04 + 1.65i)12-s + (−1.18 − 1.18i)13-s + (−1.33 + 0.456i)14-s + (0.658 + 0.912i)15-s + (1.00 − 3.87i)16-s − 7.20i·17-s + ⋯ |
L(s) = 1 | + (−0.946 + 0.323i)2-s + (−0.987 − 0.159i)3-s + (0.791 − 0.611i)4-s + (−0.205 − 0.205i)5-s + (0.985 − 0.167i)6-s + 0.377·7-s + (−0.551 + 0.834i)8-s + (0.948 + 0.315i)9-s + (0.260 + 0.128i)10-s + (−0.432 + 0.432i)11-s + (−0.878 + 0.476i)12-s + (−0.328 − 0.328i)13-s + (−0.357 + 0.122i)14-s + (0.169 + 0.235i)15-s + (0.252 − 0.967i)16-s − 1.74i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0229 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0229 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.298483 - 0.305418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.298483 - 0.305418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.33 - 0.456i)T \) |
| 3 | \( 1 + (1.70 + 0.277i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (0.459 + 0.459i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.43 - 1.43i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.18 + 1.18i)T + 13iT^{2} \) |
| 17 | \( 1 + 7.20iT - 17T^{2} \) |
| 19 | \( 1 + (2.23 - 2.23i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.540iT - 23T^{2} \) |
| 29 | \( 1 + (-4.14 + 4.14i)T - 29iT^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + (1.36 - 1.36i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.16T + 41T^{2} \) |
| 43 | \( 1 + (6.40 + 6.40i)T + 43iT^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 + (2.29 + 2.29i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.32 + 6.32i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.637 - 0.637i)T + 61iT^{2} \) |
| 67 | \( 1 + (2.14 - 2.14i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.6iT - 71T^{2} \) |
| 73 | \( 1 - 0.233iT - 73T^{2} \) |
| 79 | \( 1 - 4.93iT - 79T^{2} \) |
| 83 | \( 1 + (1.76 + 1.76i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−11.41346662708327699114187196900, −10.19331421937628120196912841689, −9.741020139539618778308540037994, −8.299151896927917441321885633137, −7.54241332670002844516330091452, −6.64340694685447752122596109222, −5.53086111762493010262971767513, −4.62154377148670086222868132357, −2.27154549511134526238434733956, −0.46735272857355362636989377010,
1.55480332965484602454214861648, 3.40725373061089349865123408019, 4.82606763309864594240060647098, 6.20180114426316569971198356624, 7.01606505660540022409097902466, 8.130249245990364495114511003990, 9.025804353311827402386056394576, 10.32330666351086980893508533922, 10.71809482419215811919824470552, 11.50515248203016879052805204049