| L(s) = 1 | + 2.26·3-s − 2.03·5-s + 2.84·7-s + 2.12·9-s + 5.94·11-s + 3.06·13-s − 4.59·15-s + 5.90·17-s + 2.58·19-s + 6.44·21-s + 1.10·23-s − 0.874·25-s − 1.98·27-s + 6.71·29-s − 6.73·31-s + 13.4·33-s − 5.78·35-s + 4.99·37-s + 6.93·39-s − 8.52·41-s − 5.24·43-s − 4.31·45-s + 10.6·47-s + 1.11·49-s + 13.3·51-s − 10.2·53-s − 12.0·55-s + ⋯ |
| L(s) = 1 | + 1.30·3-s − 0.908·5-s + 1.07·7-s + 0.707·9-s + 1.79·11-s + 0.849·13-s − 1.18·15-s + 1.43·17-s + 0.592·19-s + 1.40·21-s + 0.230·23-s − 0.174·25-s − 0.381·27-s + 1.24·29-s − 1.21·31-s + 2.34·33-s − 0.978·35-s + 0.821·37-s + 1.11·39-s − 1.33·41-s − 0.800·43-s − 0.642·45-s + 1.54·47-s + 0.159·49-s + 1.87·51-s − 1.41·53-s − 1.62·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.992469734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.992469734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.26T + 3T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 - 5.94T + 11T^{2} \) |
| 13 | \( 1 - 3.06T + 13T^{2} \) |
| 17 | \( 1 - 5.90T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 1.10T + 23T^{2} \) |
| 29 | \( 1 - 6.71T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 41 | \( 1 + 8.52T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 2.18T + 59T^{2} \) |
| 61 | \( 1 + 7.25T + 61T^{2} \) |
| 67 | \( 1 - 7.02T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 8.01T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.155495372658900225438292402841, −7.64004747953091691688655223314, −6.92175301474011934726355183599, −6.00026856279709790455650393694, −5.03547161030352897509271779548, −4.09692399321285102090722660848, −3.64441939649042506250741297390, −3.04323106057879200473367136545, −1.71351572822590095598360444283, −1.13436945614072573785820928697,
1.13436945614072573785820928697, 1.71351572822590095598360444283, 3.04323106057879200473367136545, 3.64441939649042506250741297390, 4.09692399321285102090722660848, 5.03547161030352897509271779548, 6.00026856279709790455650393694, 6.92175301474011934726355183599, 7.64004747953091691688655223314, 8.155495372658900225438292402841
