L(s) = 1 | − i·3-s + (2 − i)5-s − i·7-s + 2·9-s + 11-s − i·13-s + (−1 − 2i)15-s + 3i·17-s − 4·19-s − 21-s − 2i·23-s + (3 − 4i)25-s − 5i·27-s + 29-s + 6·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.894 − 0.447i)5-s − 0.377i·7-s + 0.666·9-s + 0.301·11-s − 0.277i·13-s + (−0.258 − 0.516i)15-s + 0.727i·17-s − 0.917·19-s − 0.218·21-s − 0.417i·23-s + (0.600 − 0.800i)25-s − 0.962i·27-s + 0.185·29-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50102 - 0.927685i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50102 - 0.927685i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 19iT - 97T^{2} \) |
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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
−10.32784410326647406069158074081, −9.954249134104325285537059379053, −8.737230499483633929453071530797, −8.029992865074042590792878729040, −6.77948403971659315174237405257, −6.27082841114926909906127331834, −5.01305810317297161823904958957, −3.98222919459956599204784297958, −2.29131801132705061393555228957, −1.16571607928687842361802773531,
1.77123885835003897391704340656, 3.06333061034806352466887240392, 4.35998212917175502672056735713, 5.30679125202027908688252562844, 6.40964565176020716426072231885, 7.11430035209296563389491696045, 8.488373296767513634131314970621, 9.380649780749393558087793542342, 9.992745442488012655709516755189, 10.70862297995796741768319571732