L(s) = 1 | − 2.58·2-s − 3-s + 4.70·4-s + 3.45·5-s + 2.58·6-s − 6.99·8-s + 9-s − 8.93·10-s − 2.74·11-s − 4.70·12-s − 1.59·13-s − 3.45·15-s + 8.71·16-s + 6.70·17-s − 2.58·18-s + 4.64·19-s + 16.2·20-s + 7.09·22-s + 0.175·23-s + 6.99·24-s + 6.91·25-s + 4.14·26-s − 27-s + 10.0·29-s + 8.93·30-s + 4.40·31-s − 8.56·32-s + ⋯ |
L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.35·4-s + 1.54·5-s + 1.05·6-s − 2.47·8-s + 0.333·9-s − 2.82·10-s − 0.826·11-s − 1.35·12-s − 0.443·13-s − 0.891·15-s + 2.17·16-s + 1.62·17-s − 0.610·18-s + 1.06·19-s + 3.63·20-s + 1.51·22-s + 0.0365·23-s + 1.42·24-s + 1.38·25-s + 0.812·26-s − 0.192·27-s + 1.87·29-s + 1.63·30-s + 0.790·31-s − 1.51·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.056195680\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.056195680\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 5 | \( 1 - 3.45T + 5T^{2} \) |
| 11 | \( 1 + 2.74T + 11T^{2} \) |
| 13 | \( 1 + 1.59T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 23 | \( 1 - 0.175T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 4.40T + 31T^{2} \) |
| 37 | \( 1 - 8.43T + 37T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 59 | \( 1 + 3.26T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 2.95T + 79T^{2} \) |
| 83 | \( 1 + 0.195T + 83T^{2} \) |
| 89 | \( 1 + 2.66T + 89T^{2} \) |
| 97 | \( 1 - 8.38T + 97T^{2} \) |
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
−7.958048674212080509868146158836, −7.66645691964898340195420054060, −6.73165282976352178251429130118, −6.13223546868165136426139826496, −5.54304803221897210786873368478, −4.81305407809019498511880115499, −2.99105633035516691937648543193, −2.51100067367004547067083587416, −1.40631428968849757579505619298, −0.817101094238779962940581018931,
0.817101094238779962940581018931, 1.40631428968849757579505619298, 2.51100067367004547067083587416, 2.99105633035516691937648543193, 4.81305407809019498511880115499, 5.54304803221897210786873368478, 6.13223546868165136426139826496, 6.73165282976352178251429130118, 7.66645691964898340195420054060, 7.958048674212080509868146158836