| L(s) = 1 | + 3-s − 0.900·5-s + 0.226·7-s + 9-s − 3.23·11-s − 6.29·13-s − 0.900·15-s + 0.102·17-s + 6.74·19-s + 0.226·21-s + 23-s − 4.18·25-s + 27-s − 29-s − 0.708·31-s − 3.23·33-s − 0.203·35-s + 1.94·37-s − 6.29·39-s − 3.13·41-s + 4.38·43-s − 0.900·45-s + 6.47·47-s − 6.94·49-s + 0.102·51-s − 7.91·53-s + 2.91·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.402·5-s + 0.0856·7-s + 0.333·9-s − 0.975·11-s − 1.74·13-s − 0.232·15-s + 0.0249·17-s + 1.54·19-s + 0.0494·21-s + 0.208·23-s − 0.837·25-s + 0.192·27-s − 0.185·29-s − 0.127·31-s − 0.563·33-s − 0.0344·35-s + 0.320·37-s − 1.00·39-s − 0.489·41-s + 0.668·43-s − 0.134·45-s + 0.944·47-s − 0.992·49-s + 0.0143·51-s − 1.08·53-s + 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.739951555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.739951555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 0.900T + 5T^{2} \) |
| 7 | \( 1 - 0.226T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 - 0.102T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 31 | \( 1 + 0.708T + 31T^{2} \) |
| 37 | \( 1 - 1.94T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 - 4.38T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 7.91T + 53T^{2} \) |
| 59 | \( 1 - 8.52T + 59T^{2} \) |
| 61 | \( 1 - 7.66T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.53T + 71T^{2} \) |
| 73 | \( 1 - 4.34T + 73T^{2} \) |
| 79 | \( 1 + 7.86T + 79T^{2} \) |
| 83 | \( 1 - 6.62T + 83T^{2} \) |
| 89 | \( 1 - 0.363T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
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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
−7.77417701399740543776087734020, −7.38796315551081228263755760893, −6.69241462000932551105589190236, −5.44626861617862379410342948219, −5.14345939368663740300830905804, −4.26659661829459597939274091572, −3.39830747312928594124035579413, −2.68353733895812561124978567695, −1.99722369545435995307414707085, −0.60932525063701561510798155696,
0.60932525063701561510798155696, 1.99722369545435995307414707085, 2.68353733895812561124978567695, 3.39830747312928594124035579413, 4.26659661829459597939274091572, 5.14345939368663740300830905804, 5.44626861617862379410342948219, 6.69241462000932551105589190236, 7.38796315551081228263755760893, 7.77417701399740543776087734020
