| L(s) = 1 | + 0.0278·3-s − 0.122·5-s + 0.877·7-s − 2.99·9-s + 11-s + 1.68·13-s − 0.00340·15-s + 6.34·17-s − 19-s + 0.0244·21-s + 6.12·23-s − 4.98·25-s − 0.166·27-s + 3.64·29-s − 0.00340·31-s + 0.0278·33-s − 0.107·35-s − 8.21·37-s + 0.0468·39-s − 11.1·41-s − 0.0339·43-s + 0.367·45-s + 6.10·47-s − 6.22·49-s + 0.176·51-s + 7.24·53-s − 0.122·55-s + ⋯ |
| L(s) = 1 | + 0.0160·3-s − 0.0547·5-s + 0.331·7-s − 0.999·9-s + 0.301·11-s + 0.467·13-s − 0.000879·15-s + 1.53·17-s − 0.229·19-s + 0.00533·21-s + 1.27·23-s − 0.997·25-s − 0.0321·27-s + 0.676·29-s − 0.000611·31-s + 0.00484·33-s − 0.0181·35-s − 1.34·37-s + 0.00750·39-s − 1.74·41-s − 0.00517·43-s + 0.0547·45-s + 0.891·47-s − 0.889·49-s + 0.0247·51-s + 0.995·53-s − 0.0165·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.883427453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.883427453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0278T + 3T^{2} \) |
| 5 | \( 1 + 0.122T + 5T^{2} \) |
| 7 | \( 1 - 0.877T + 7T^{2} \) |
| 13 | \( 1 - 1.68T + 13T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 23 | \( 1 - 6.12T + 23T^{2} \) |
| 29 | \( 1 - 3.64T + 29T^{2} \) |
| 31 | \( 1 + 0.00340T + 31T^{2} \) |
| 37 | \( 1 + 8.21T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 0.0339T + 43T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 - 7.24T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 4.05T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 + 0.0136T + 71T^{2} \) |
| 73 | \( 1 - 3.73T + 73T^{2} \) |
| 79 | \( 1 - 0.356T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−8.534894117364714417986308700457, −8.048810452510903664341350242739, −7.13839353313720723913643066359, −6.36333455574032149992190114945, −5.48402374491544367836492081649, −5.01588857882208054029501169665, −3.72734332934737422427595096204, −3.19353872488499630600546846899, −2.01236809741968421995085304345, −0.833165468390809403412805853022,
0.833165468390809403412805853022, 2.01236809741968421995085304345, 3.19353872488499630600546846899, 3.72734332934737422427595096204, 5.01588857882208054029501169665, 5.48402374491544367836492081649, 6.36333455574032149992190114945, 7.13839353313720723913643066359, 8.048810452510903664341350242739, 8.534894117364714417986308700457
