L(s) = 1 | − 3-s + 2.55·5-s − 3.69·7-s + 9-s + 5.01·11-s + 5.57·13-s − 2.55·15-s − 1.33·17-s + 8.44·19-s + 3.69·21-s + 6.52·23-s + 1.53·25-s − 27-s − 10.1·29-s + 2.82·31-s − 5.01·33-s − 9.45·35-s + 6.51·37-s − 5.57·39-s + 7.07·41-s − 4.12·43-s + 2.55·45-s + 13.4·47-s + 6.68·49-s + 1.33·51-s + 4.54·53-s + 12.8·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.14·5-s − 1.39·7-s + 0.333·9-s + 1.51·11-s + 1.54·13-s − 0.659·15-s − 0.324·17-s + 1.93·19-s + 0.807·21-s + 1.36·23-s + 0.306·25-s − 0.192·27-s − 1.87·29-s + 0.507·31-s − 0.873·33-s − 1.59·35-s + 1.07·37-s − 0.892·39-s + 1.10·41-s − 0.629·43-s + 0.380·45-s + 1.95·47-s + 0.955·49-s + 0.187·51-s + 0.623·53-s + 1.72·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.460152935\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.460152935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 2.55T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 - 5.01T + 11T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 8.44T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 + 4.12T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 - 4.54T + 53T^{2} \) |
| 59 | \( 1 + 8.25T + 59T^{2} \) |
| 61 | \( 1 + 5.80T + 61T^{2} \) |
| 67 | \( 1 - 7.81T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 + 6.04T + 73T^{2} \) |
| 79 | \( 1 + 6.04T + 79T^{2} \) |
| 83 | \( 1 + 3.09T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 0.668T + 97T^{2} \) |
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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
−7.55723360478386523406722588140, −6.95517900845283900378222893033, −6.21460192846846116200325970864, −5.97681071015086099794700087596, −5.35080389306362122910055775259, −4.15259188904032717973307746257, −3.54284956108390413078263305609, −2.76975855024723041253531825101, −1.47559796828296859477509443992, −0.905430025623087782128210164746,
0.905430025623087782128210164746, 1.47559796828296859477509443992, 2.76975855024723041253531825101, 3.54284956108390413078263305609, 4.15259188904032717973307746257, 5.35080389306362122910055775259, 5.97681071015086099794700087596, 6.21460192846846116200325970864, 6.95517900845283900378222893033, 7.55723360478386523406722588140