L(s) = 1 | + (0.309 + 0.535i)2-s + (−0.309 − 0.535i)3-s + (0.309 − 0.535i)4-s + (0.190 − 0.330i)6-s + (−0.809 + 1.40i)7-s + 0.999·8-s + (0.309 − 0.535i)9-s + (0.5 + 0.866i)11-s − 0.381·12-s − 14-s + 0.381·18-s + (0.309 − 0.535i)19-s + 21-s + (−0.309 + 0.535i)22-s + (0.809 + 1.40i)23-s + (−0.309 − 0.535i)24-s + ⋯ |
L(s) = 1 | + (0.309 + 0.535i)2-s + (−0.309 − 0.535i)3-s + (0.309 − 0.535i)4-s + (0.190 − 0.330i)6-s + (−0.809 + 1.40i)7-s + 0.999·8-s + (0.309 − 0.535i)9-s + (0.5 + 0.866i)11-s − 0.381·12-s − 14-s + 0.381·18-s + (0.309 − 0.535i)19-s + 21-s + (−0.309 + 0.535i)22-s + (0.809 + 1.40i)23-s + (−0.309 − 0.535i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.381345118\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381345118\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.61T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.381981748916703566313029652126, −8.858960169227901867166573903788, −7.44892659256378597463855215934, −6.91789388936982228852827860508, −6.36562870835256555025457009635, −5.56225393262291710788002689167, −4.96060050314026756305114350412, −3.63255972702343510523761362683, −2.42655582557872478330818560558, −1.35514688192140236055118122844,
1.20626941916860746336992408420, 2.79834940735570818638491562373, 3.58571279108148243151393541180, 4.25849949411059740883304861699, 5.01803144777260685952954823991, 6.38495362338698665169902262518, 6.94722450545858766786650154676, 7.76787933143495944653112452260, 8.600917816504514829540414896601, 9.688242645501371632220770095548