| L(s) = 1 | − 2.67·3-s + 3.55·5-s + 1.65·7-s + 4.14·9-s + 1.12·11-s − 2.27·13-s − 9.50·15-s − 3.07·17-s + 19-s − 4.43·21-s + 4.46·23-s + 7.62·25-s − 3.07·27-s − 3.82·29-s − 4.45·31-s − 2.99·33-s + 5.89·35-s − 11.5·37-s + 6.08·39-s − 9.46·41-s + 4.63·43-s + 14.7·45-s − 2.15·47-s − 4.25·49-s + 8.22·51-s − 7.35·53-s + 3.98·55-s + ⋯ |
| L(s) = 1 | − 1.54·3-s + 1.58·5-s + 0.626·7-s + 1.38·9-s + 0.337·11-s − 0.631·13-s − 2.45·15-s − 0.746·17-s + 0.229·19-s − 0.967·21-s + 0.931·23-s + 1.52·25-s − 0.591·27-s − 0.710·29-s − 0.799·31-s − 0.521·33-s + 0.995·35-s − 1.90·37-s + 0.974·39-s − 1.47·41-s + 0.706·43-s + 2.19·45-s − 0.314·47-s − 0.607·49-s + 1.15·51-s − 1.01·53-s + 0.536·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 2.67T + 3T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 31 | \( 1 + 4.45T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 - 4.63T + 43T^{2} \) |
| 47 | \( 1 + 2.15T + 47T^{2} \) |
| 53 | \( 1 + 7.35T + 53T^{2} \) |
| 59 | \( 1 + 3.21T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 4.05T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.30829926880428598157118197613, −6.90587787950628697290735179008, −6.14197393301011575989894682437, −5.57432325191242305322496548694, −5.04087462230671361751656129528, −4.52474824974650785065179597024, −3.15105131440210899197791156282, −1.91154269231952280543856814494, −1.43228508078177691804410637923, 0,
1.43228508078177691804410637923, 1.91154269231952280543856814494, 3.15105131440210899197791156282, 4.52474824974650785065179597024, 5.04087462230671361751656129528, 5.57432325191242305322496548694, 6.14197393301011575989894682437, 6.90587787950628697290735179008, 7.30829926880428598157118197613
