L(s) = 1 | + (0.669 − 0.743i)5-s − 1.95·7-s + (0.309 + 0.951i)9-s + (−0.604 + 1.86i)11-s + (−1.47 + 1.07i)17-s + (−0.809 + 0.587i)19-s + (0.190 − 0.587i)23-s + (−0.104 − 0.994i)25-s + (−1.30 + 1.45i)35-s − 0.209·43-s + (0.913 + 0.406i)45-s + (0.809 + 0.587i)47-s + 2.82·49-s + (0.978 + 1.69i)55-s + (−0.0646 + 0.198i)61-s + ⋯ |
L(s) = 1 | + (0.669 − 0.743i)5-s − 1.95·7-s + (0.309 + 0.951i)9-s + (−0.604 + 1.86i)11-s + (−1.47 + 1.07i)17-s + (−0.809 + 0.587i)19-s + (0.190 − 0.587i)23-s + (−0.104 − 0.994i)25-s + (−1.30 + 1.45i)35-s − 0.209·43-s + (0.913 + 0.406i)45-s + (0.809 + 0.587i)47-s + 2.82·49-s + (0.978 + 1.69i)55-s + (−0.0646 + 0.198i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6157253839\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6157253839\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.669 + 0.743i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
good | 3 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + 1.95T + T^{2} \) |
| 11 | \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 0.209T + T^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.684804790140483080793694469653, −9.018690215250002404408444936851, −8.173953164508901407961936142432, −7.11018630610474474147004168876, −6.51101058406748575614743852181, −5.73235625159288982913902835417, −4.64082405902181401531915526424, −4.10342165060886491190830641064, −2.53543804531689072451829621519, −1.93194513664864831252879003522,
0.40962753579880184930645718525, 2.56019610223348062616543685430, 3.11594554935862544148515280692, 3.88214336155628197450283314049, 5.39926521858662689300789627022, 6.28340224627399444358030428912, 6.54475411905185056114624808950, 7.27958803898752901507456862451, 8.752937483824783633056489339447, 9.190887020284336321273738549700