L(s) = 1 | + (1.56 − 0.900i)2-s + (1.12 − 1.94i)4-s + (−0.866 − 0.5i)7-s − 2.24i·8-s + (−0.5 + 0.866i)9-s + (1.07 − 0.623i)11-s − 1.80·14-s + (−0.900 − 1.56i)16-s + 1.80i·18-s + (1.12 − 1.94i)22-s + (−0.222 − 0.385i)23-s − 25-s + (−1.94 + 1.12i)28-s + (0.900 + 1.56i)29-s + (−0.866 − 0.500i)32-s + ⋯ |
L(s) = 1 | + (1.56 − 0.900i)2-s + (1.12 − 1.94i)4-s + (−0.866 − 0.5i)7-s − 2.24i·8-s + (−0.5 + 0.866i)9-s + (1.07 − 0.623i)11-s − 1.80·14-s + (−0.900 − 1.56i)16-s + 1.80i·18-s + (1.12 − 1.94i)22-s + (−0.222 − 0.385i)23-s − 25-s + (−1.94 + 1.12i)28-s + (0.900 + 1.56i)29-s + (−0.866 − 0.500i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0502 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0502 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.259233437\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.259233437\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 0.445T + T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.385 + 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 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
−10.14712218980951444014893653361, −9.214283085125895941472573255075, −8.091914249678973365617626000867, −6.75570992410048535535773027496, −6.23088715286458713420184627815, −5.30266348973670830105220811097, −4.40351161863083473325251855968, −3.51609540835881232749599847832, −2.83340053426143836574248479789, −1.50361970826528540827351689115,
2.37704253381719336701684406717, 3.54053353275317019788777705355, 4.02039784765041429902795604060, 5.20646043187583632361579894760, 6.12718972223620377390796361162, 6.46056485089736677950921781566, 7.29561611769595736008861179962, 8.344905381722920352869146329962, 9.268386728702999664991733709872, 10.00912549386231903036586926334