L(s) = 1 | + (0.382 − 0.923i)2-s + (0.541 + 0.541i)3-s + (−0.707 − 0.707i)4-s + (0.707 − 0.292i)6-s + (−0.923 + 0.382i)8-s − 0.414i·9-s − 1.41i·11-s − 0.765i·12-s + (0.541 − 0.541i)13-s + i·16-s + (−0.382 − 0.158i)18-s − 19-s + (−1.30 − 0.541i)22-s + (−0.707 − 0.292i)24-s + (−0.292 − 0.707i)26-s + (0.765 − 0.765i)27-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)2-s + (0.541 + 0.541i)3-s + (−0.707 − 0.707i)4-s + (0.707 − 0.292i)6-s + (−0.923 + 0.382i)8-s − 0.414i·9-s − 1.41i·11-s − 0.765i·12-s + (0.541 − 0.541i)13-s + i·16-s + (−0.382 − 0.158i)18-s − 19-s + (−1.30 − 0.541i)22-s + (−0.707 − 0.292i)24-s + (−0.292 − 0.707i)26-s + (0.765 − 0.765i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.462008315\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.462008315\) |
\(L(1)\) |
|
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 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
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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
−9.144477763631675689719249578117, −8.673681943826394602515327545581, −8.049533285787338494738502927912, −6.43967725205166585907823980429, −5.92520657437734001119676129784, −4.87269846583311180921227946875, −3.90296581463178341189035900458, −3.34539004702414057448007706418, −2.50562145621444734270951910066, −0.948020675775657191708659355364,
1.82981931649332210426262911042, 2.82521972678388321272019008533, 4.18349282811218865180569265000, 4.62328336452092402814194591423, 5.80631318650078260808157781364, 6.55966938591925544238696870226, 7.44932511482799549510543305668, 7.71249890605844308698971627928, 8.779650969188698369497979499097, 9.199301876710615110911715381614