L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s − 7-s + (−0.707 − 0.707i)8-s + (1.41 − 1.41i)11-s + (−0.707 + 0.707i)14-s − 1.00·16-s − 2.00i·22-s − 1.41i·23-s + i·25-s + 1.00i·28-s + (−0.707 + 0.707i)32-s + (−1 + i)37-s + (1 + i)43-s + (−1.41 − 1.41i)44-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s − 7-s + (−0.707 − 0.707i)8-s + (1.41 − 1.41i)11-s + (−0.707 + 0.707i)14-s − 1.00·16-s − 2.00i·22-s − 1.41i·23-s + i·25-s + 1.00i·28-s + (−0.707 + 0.707i)32-s + (−1 + i)37-s + (1 + i)43-s + (−1.41 − 1.41i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.342125338\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.342125338\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.10231523554190349916993408846, −9.129786693334098306597092087581, −8.704060716371164168274111093784, −7.06165034340317766576933209663, −6.28218075200214905304495097913, −5.72864841652604421237337762113, −4.38899983866581797258319799797, −3.54904352836900085482281858298, −2.77809476993158973441013532046, −1.11622772767520089984220746573,
2.14637640305381176412135535574, 3.56523649172718758753114564908, 4.15127228800176227740184536227, 5.29422179571932956422919300205, 6.25622088453901574577005321352, 6.95264024343878514512793434508, 7.51893891723800778170102832668, 8.821896515855332544558757475340, 9.394545479879551984689012205427, 10.23286915262435904981989699175