L(s) = 1 | − 2.65·3-s − 0.749·5-s − 1.02·7-s + 4.04·9-s + 0.361·11-s − 5.83·13-s + 1.98·15-s + 0.216·17-s − 6.45·19-s + 2.71·21-s + 2.08·23-s − 4.43·25-s − 2.77·27-s − 5.90·29-s + 3.13·31-s − 0.959·33-s + 0.766·35-s − 7.09·37-s + 15.4·39-s + 3.00·41-s − 10.4·43-s − 3.02·45-s + 10.7·47-s − 5.95·49-s − 0.575·51-s + 9.95·53-s − 0.270·55-s + ⋯ |
L(s) = 1 | − 1.53·3-s − 0.335·5-s − 0.386·7-s + 1.34·9-s + 0.108·11-s − 1.61·13-s + 0.513·15-s + 0.0525·17-s − 1.48·19-s + 0.592·21-s + 0.434·23-s − 0.887·25-s − 0.533·27-s − 1.09·29-s + 0.562·31-s − 0.166·33-s + 0.129·35-s − 1.16·37-s + 2.48·39-s + 0.469·41-s − 1.59·43-s − 0.451·45-s + 1.57·47-s − 0.850·49-s − 0.0805·51-s + 1.36·53-s − 0.0365·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1720197182\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1720197182\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 + 2.65T + 3T^{2} \) |
| 5 | \( 1 + 0.749T + 5T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 11 | \( 1 - 0.361T + 11T^{2} \) |
| 13 | \( 1 + 5.83T + 13T^{2} \) |
| 17 | \( 1 - 0.216T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 - 3.13T + 31T^{2} \) |
| 37 | \( 1 + 7.09T + 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 9.95T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 - 8.86T + 71T^{2} \) |
| 73 | \( 1 + 3.64T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 1.31T + 83T^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.87328625857673781893746033784, −7.14362311963987557196971795240, −6.66401284486457411114656878916, −5.90199491801352755610898710634, −5.27435452674441660823520725840, −4.57806966429185399088090990432, −3.92705720081859689498392748512, −2.73366612767364393480967876701, −1.68733966763394626020925007863, −0.22917733966975928595688408250,
0.22917733966975928595688408250, 1.68733966763394626020925007863, 2.73366612767364393480967876701, 3.92705720081859689498392748512, 4.57806966429185399088090990432, 5.27435452674441660823520725840, 5.90199491801352755610898710634, 6.66401284486457411114656878916, 7.14362311963987557196971795240, 7.87328625857673781893746033784