| L(s) = 1 | + 3-s + 3·5-s − 7-s + 9-s + 11-s + 13-s + 3·15-s − 1.58·17-s − 2.58·19-s − 21-s − 3.58·23-s + 4·25-s + 27-s + 10.1·29-s + 5.58·31-s + 33-s − 3·35-s + 37-s + 39-s + 7.16·41-s + 7.58·43-s + 3·45-s − 10.5·47-s + 49-s − 1.58·51-s − 0.417·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.774·15-s − 0.383·17-s − 0.592·19-s − 0.218·21-s − 0.747·23-s + 0.800·25-s + 0.192·27-s + 1.88·29-s + 1.00·31-s + 0.174·33-s − 0.507·35-s + 0.164·37-s + 0.160·39-s + 1.11·41-s + 1.15·43-s + 0.447·45-s − 1.54·47-s + 0.142·49-s − 0.221·51-s − 0.0573·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.116720402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.116720402\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3T + 5T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 7.16T + 41T^{2} \) |
| 43 | \( 1 - 7.58T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.417T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 0.582T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 2.41T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.500689446833055001099449387147, −8.042568480195725916233526167866, −6.74534472074773064211733407631, −6.44149624567824050826581296683, −5.66992594967210592452856750986, −4.67532184049123285660327163647, −3.87537894885409196915233245817, −2.73441290422602185310196572047, −2.18253588002634172535927680470, −1.05219458621003520936746611663,
1.05219458621003520936746611663, 2.18253588002634172535927680470, 2.73441290422602185310196572047, 3.87537894885409196915233245817, 4.67532184049123285660327163647, 5.66992594967210592452856750986, 6.44149624567824050826581296683, 6.74534472074773064211733407631, 8.042568480195725916233526167866, 8.500689446833055001099449387147
