| L(s) = 1 | − 2-s − 0.347·3-s + 4-s + 0.347·6-s − 8-s − 0.879·9-s + 1.53·11-s − 0.347·12-s + 16-s − 17-s + 0.879·18-s − 1.53·22-s + 0.347·24-s + 25-s + 0.652·27-s − 32-s − 0.532·33-s + 34-s − 0.879·36-s + 1.87·41-s − 43-s + 1.53·44-s − 0.347·48-s + 49-s − 50-s + 0.347·51-s − 0.652·54-s + ⋯ |
| L(s) = 1 | − 2-s − 0.347·3-s + 4-s + 0.347·6-s − 8-s − 0.879·9-s + 1.53·11-s − 0.347·12-s + 16-s − 17-s + 0.879·18-s − 1.53·22-s + 0.347·24-s + 25-s + 0.652·27-s − 32-s − 0.532·33-s + 34-s − 0.879·36-s + 1.87·41-s − 43-s + 1.53·44-s − 0.347·48-s + 49-s − 50-s + 0.347·51-s − 0.652·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6881023955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6881023955\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.347T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.53T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.87T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.53T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.87T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 - 1.87T + T^{2} \) |
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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
−8.993678517698826143931529905819, −8.429250142831578527476831727452, −7.48526290844213018085038823308, −6.57370698852761472192408807411, −6.30386950254554263872415327435, −5.28674696023220523436484581613, −4.16207856650570433711671946561, −3.11878518588072946754418955709, −2.12604960266300348604347603922, −0.898999948754934995127462865547,
0.898999948754934995127462865547, 2.12604960266300348604347603922, 3.11878518588072946754418955709, 4.16207856650570433711671946561, 5.28674696023220523436484581613, 6.30386950254554263872415327435, 6.57370698852761472192408807411, 7.48526290844213018085038823308, 8.429250142831578527476831727452, 8.993678517698826143931529905819
