| L(s) = 1 | − 2.51i·2-s − 4.34·4-s + (−2.08 + 0.814i)5-s + 4.08i·7-s + 5.91i·8-s + (2.05 + 5.24i)10-s − 1.07·11-s − 1.43i·13-s + 10.2·14-s + 6.20·16-s − 6.88i·17-s + 7.42·19-s + (9.05 − 3.54i)20-s + 2.71i·22-s − 1.01i·23-s + ⋯ |
| L(s) = 1 | − 1.78i·2-s − 2.17·4-s + (−0.931 + 0.364i)5-s + 1.54i·7-s + 2.09i·8-s + (0.649 + 1.65i)10-s − 0.325·11-s − 0.397i·13-s + 2.74·14-s + 1.55·16-s − 1.67i·17-s + 1.70·19-s + (2.02 − 0.791i)20-s + 0.579i·22-s − 0.212i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9159825093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9159825093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.08 - 0.814i)T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51iT - 2T^{2} \) |
| 7 | \( 1 - 4.08iT - 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 + 1.43iT - 13T^{2} \) |
| 17 | \( 1 + 6.88iT - 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 + 1.01iT - 23T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 3.02T + 41T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 + 6.88iT - 47T^{2} \) |
| 53 | \( 1 + 7.61iT - 53T^{2} \) |
| 59 | \( 1 - 9.45T + 59T^{2} \) |
| 61 | \( 1 + 0.265T + 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 3.54iT - 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 + 0.615iT - 83T^{2} \) |
| 89 | \( 1 + 2.11T + 89T^{2} \) |
| 97 | \( 1 + 1.52iT - 97T^{2} \) |
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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
−9.456493729127356869746366715485, −8.819116512791543564314217671813, −7.983581619846224298388425291085, −6.98597325792757273185133884237, −5.35683279495282860745992699326, −4.98489961500663023119837333761, −3.54338749256540867908572321042, −2.99845054150616283521683998093, −2.16636464431100601227427607249, −0.50508313319113510134729190445,
1.00137814177975605781126531849, 3.70397918863717092373871326478, 4.11074820276289287942859980744, 5.09112242926666778956215673396, 5.95937368050826339435452966578, 7.00331982023377693267108251227, 7.59651455451584659647740476989, 7.88685538430706166328959137706, 8.914964283196376880583561168517, 9.648500507442993924851423753318
