L(s) = 1 | + (0.382 + 0.923i)2-s + (1.30 + 1.30i)3-s + (−0.707 + 0.707i)4-s + (−0.707 + 1.70i)6-s + (−0.923 − 0.382i)8-s + 2.41i·9-s − 1.41i·11-s − 1.84·12-s + (−1.30 + 1.30i)13-s − i·16-s + (−2.23 + 0.923i)18-s + 19-s + (1.30 − 0.541i)22-s + (−0.707 − 1.70i)24-s + (−1.70 − 0.707i)26-s + (−1.84 + 1.84i)27-s + ⋯ |
L(s) = 1 | + (0.382 + 0.923i)2-s + (1.30 + 1.30i)3-s + (−0.707 + 0.707i)4-s + (−0.707 + 1.70i)6-s + (−0.923 − 0.382i)8-s + 2.41i·9-s − 1.41i·11-s − 1.84·12-s + (−1.30 + 1.30i)13-s − i·16-s + (−2.23 + 0.923i)18-s + 19-s + (1.30 − 0.541i)22-s + (−0.707 − 1.70i)24-s + (−1.70 − 0.707i)26-s + (−1.84 + 1.84i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.832074668\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.832074668\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
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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
−9.570907780152411265683880719247, −8.879785385300731468245046093483, −8.322243184094183552609890507362, −7.57652977702784481390779753621, −6.71756971969650903539382221091, −5.48743641422724066387026333668, −4.85836513371519617267666041308, −4.01514808232187882780723276098, −3.32533611319583531106423311152, −2.46411117762019502850953126992,
1.09114076169499425814473045388, 2.26597779040219881941539119828, 2.71710528807948341605507902848, 3.68979287640382563508939471400, 4.80401917601031926557068542042, 5.75862247154546065618286476282, 6.97014170306568899729369432647, 7.54083111431792685847880665059, 8.188564413639549159227791267304, 9.226794020253670580631519748069