| L(s) = 1 | − 0.146·3-s + 2.34·5-s + 3.83·7-s − 2.97·9-s + 3.34·11-s + 6.17·13-s − 0.342·15-s + 5.19·17-s − 0.560·21-s + 2.34·23-s + 0.489·25-s + 0.875·27-s + 0.0501·29-s − 3.43·31-s − 0.489·33-s + 8.97·35-s − 5.43·37-s − 0.903·39-s − 7.29·41-s − 8.86·43-s − 6.97·45-s + 10.7·47-s + 7.68·49-s − 0.760·51-s − 3.19·53-s + 7.83·55-s + 3.85·59-s + ⋯ |
| L(s) = 1 | − 0.0845·3-s + 1.04·5-s + 1.44·7-s − 0.992·9-s + 1.00·11-s + 1.71·13-s − 0.0885·15-s + 1.26·17-s − 0.122·21-s + 0.488·23-s + 0.0978·25-s + 0.168·27-s + 0.00932·29-s − 0.617·31-s − 0.0851·33-s + 1.51·35-s − 0.894·37-s − 0.144·39-s − 1.13·41-s − 1.35·43-s − 1.04·45-s + 1.56·47-s + 1.09·49-s − 0.106·51-s − 0.439·53-s + 1.05·55-s + 0.501·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.944193708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.944193708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.146T + 3T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 6.17T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 - 0.0501T + 29T^{2} \) |
| 31 | \( 1 + 3.43T + 31T^{2} \) |
| 37 | \( 1 + 5.43T + 37T^{2} \) |
| 41 | \( 1 + 7.29T + 41T^{2} \) |
| 43 | \( 1 + 8.86T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 - 3.85T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 1.48T + 71T^{2} \) |
| 73 | \( 1 + 7.68T + 73T^{2} \) |
| 79 | \( 1 - 0.175T + 79T^{2} \) |
| 83 | \( 1 + 7.00T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−8.617781758286401411401147026807, −8.338553062832413241488433992246, −7.24251204551751573295572654712, −6.28863265323901321591171644067, −5.65081038553404361773733689542, −5.17394061017284096222034441431, −3.98147961842964076630950390066, −3.13011180503176896545404690110, −1.76512126888006407099374772359, −1.25755954460316657228148922143,
1.25755954460316657228148922143, 1.76512126888006407099374772359, 3.13011180503176896545404690110, 3.98147961842964076630950390066, 5.17394061017284096222034441431, 5.65081038553404361773733689542, 6.28863265323901321591171644067, 7.24251204551751573295572654712, 8.338553062832413241488433992246, 8.617781758286401411401147026807
