L(s) = 1 | + (0.995 − 0.0922i)2-s + (0.982 − 0.183i)4-s + (0.850 + 1.52i)5-s + (0.961 − 0.273i)8-s + (−0.895 − 0.445i)9-s + (0.987 + 1.44i)10-s + (−0.489 − 1.72i)13-s + (0.932 − 0.361i)16-s + (−0.932 + 0.361i)17-s + (−0.932 − 0.361i)18-s + (1.11 + 1.34i)20-s + (−1.08 + 1.74i)25-s + (−0.646 − 1.66i)26-s + (−1.07 + 1.56i)29-s + (0.895 − 0.445i)32-s + ⋯ |
L(s) = 1 | + (0.995 − 0.0922i)2-s + (0.982 − 0.183i)4-s + (0.850 + 1.52i)5-s + (0.961 − 0.273i)8-s + (−0.895 − 0.445i)9-s + (0.987 + 1.44i)10-s + (−0.489 − 1.72i)13-s + (0.932 − 0.361i)16-s + (−0.932 + 0.361i)17-s + (−0.932 − 0.361i)18-s + (1.11 + 1.34i)20-s + (−1.08 + 1.74i)25-s + (−0.646 − 1.66i)26-s + (−1.07 + 1.56i)29-s + (0.895 − 0.445i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.025260282\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.025260282\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.995 + 0.0922i)T \) |
| 17 | \( 1 + (0.932 - 0.361i)T \) |
good | 3 | \( 1 + (0.895 + 0.445i)T^{2} \) |
| 5 | \( 1 + (-0.850 - 1.52i)T + (-0.526 + 0.850i)T^{2} \) |
| 7 | \( 1 + (-0.798 + 0.602i)T^{2} \) |
| 11 | \( 1 + (-0.361 - 0.932i)T^{2} \) |
| 13 | \( 1 + (0.489 + 1.72i)T + (-0.850 + 0.526i)T^{2} \) |
| 19 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
| 23 | \( 1 + (-0.798 + 0.602i)T^{2} \) |
| 29 | \( 1 + (1.07 - 1.56i)T + (-0.361 - 0.932i)T^{2} \) |
| 31 | \( 1 + (-0.526 - 0.850i)T^{2} \) |
| 37 | \( 1 + (-1.27 + 0.0590i)T + (0.995 - 0.0922i)T^{2} \) |
| 41 | \( 1 + (0.947 + 0.222i)T + (0.895 + 0.445i)T^{2} \) |
| 43 | \( 1 + (0.739 + 0.673i)T^{2} \) |
| 47 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 53 | \( 1 + (1.42 + 0.711i)T + (0.602 + 0.798i)T^{2} \) |
| 59 | \( 1 + (-0.273 - 0.961i)T^{2} \) |
| 61 | \( 1 + (-0.268 + 1.92i)T + (-0.961 - 0.273i)T^{2} \) |
| 67 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 71 | \( 1 + (0.798 - 0.602i)T^{2} \) |
| 73 | \( 1 + (0.256 - 0.581i)T + (-0.673 - 0.739i)T^{2} \) |
| 79 | \( 1 + (-0.183 + 0.982i)T^{2} \) |
| 83 | \( 1 + (0.445 + 0.895i)T^{2} \) |
| 89 | \( 1 + (0.544 - 1.91i)T + (-0.850 - 0.526i)T^{2} \) |
| 97 | \( 1 + (0.524 + 1.56i)T + (-0.798 + 0.602i)T^{2} \) |
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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
−10.30069927467995364663396003172, −9.517184834404302228360389654043, −8.202175904671347451888333157077, −7.21246947744655740180898756779, −6.51495877804544461148032118156, −5.79877159384346335065276568482, −5.14907783594647636121208698373, −3.53677670260518655986044809428, −2.97885818212187197857339958169, −2.10460164519940506045807827662,
1.77107474032880831894118929692, 2.53223799197842162879504569454, 4.30342108250513758173591076686, 4.66284769051343612368848308010, 5.67084728579025526298956271761, 6.23017182857803668437695885033, 7.35496582056929114683320473129, 8.368489671352762148932275120096, 9.146072860032920200479919877885, 9.776439517479769880224974162322