| L(s) = 1 | − i·2-s + (1.64 + 0.550i)3-s − 4-s + 1.99·5-s + (0.550 − 1.64i)6-s + (2.64 + 0.0703i)7-s + i·8-s + (2.39 + 1.80i)9-s − 1.99i·10-s − 1.51i·11-s + (−1.64 − 0.550i)12-s + 5.17i·13-s + (0.0703 − 2.64i)14-s + (3.27 + 1.09i)15-s + 16-s − 1.10·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.948 + 0.317i)3-s − 0.5·4-s + 0.890·5-s + (0.224 − 0.670i)6-s + (0.999 + 0.0265i)7-s + 0.353i·8-s + (0.797 + 0.602i)9-s − 0.629i·10-s − 0.455i·11-s + (−0.474 − 0.158i)12-s + 1.43i·13-s + (0.0187 − 0.706i)14-s + (0.844 + 0.283i)15-s + 0.250·16-s − 0.267·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.64496 - 0.467758i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.64496 - 0.467758i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.64 - 0.550i)T \) |
| 7 | \( 1 + (-2.64 - 0.0703i)T \) |
| 23 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.99T + 5T^{2} \) |
| 11 | \( 1 + 1.51iT - 11T^{2} \) |
| 13 | \( 1 - 5.17iT - 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 - 0.372iT - 19T^{2} \) |
| 29 | \( 1 + 1.32iT - 29T^{2} \) |
| 31 | \( 1 + 2.16iT - 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 6.00T + 47T^{2} \) |
| 53 | \( 1 + 3.58iT - 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 5.71iT - 61T^{2} \) |
| 67 | \( 1 + 6.62T + 67T^{2} \) |
| 71 | \( 1 + 13.6iT - 71T^{2} \) |
| 73 | \( 1 + 9.75iT - 73T^{2} \) |
| 79 | \( 1 + 3.85T + 79T^{2} \) |
| 83 | \( 1 + 0.708T + 83T^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 + 1.06iT - 97T^{2} \) |
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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
−9.976805040504531039681304017158, −9.076534484258516019612731812911, −8.694266289415944511609626999331, −7.71669504370010722151406832003, −6.60452666126193257254767365883, −5.33342172742332816293745562896, −4.49163430883825797441143113593, −3.57384094274819333606916060827, −2.21290866355554596304761304020, −1.69855019169274234892245727346,
1.38678350118753452015480333670, 2.51371346245627826026307324318, 3.79974736973505837280240745946, 5.01317956475488068242531127088, 5.71212972818098118448939211730, 6.90524882539372811607056355786, 7.55075482419110833025962224143, 8.419035436348345803771343860718, 8.914512424162860982840742186547, 10.01940230899697603217081023487
