| L(s) = 1 | − 1.41·5-s − 7.07·13-s + 8·17-s − 2.99·25-s − 4.24·29-s + 9.89·37-s + 8·41-s − 7·49-s − 12.7·53-s − 15.5·61-s + 10.0·65-s + 6·73-s − 11.3·85-s + 10·89-s − 8·97-s + 12.7·101-s + 18.3·109-s − 14·113-s + ⋯ |
| L(s) = 1 | − 0.632·5-s − 1.96·13-s + 1.94·17-s − 0.599·25-s − 0.787·29-s + 1.62·37-s + 1.24·41-s − 49-s − 1.74·53-s − 1.99·61-s + 1.24·65-s + 0.702·73-s − 1.22·85-s + 1.05·89-s − 0.812·97-s + 1.26·101-s + 1.76·109-s − 1.31·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.263513826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263513826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 - 8T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.81046155313820667971289187945, −7.34024456024990360313689940413, −6.30998251773146079328586458311, −5.63723162264792662564034588512, −4.89070274351756448540657880811, −4.29611386479669596781442295531, −3.36991511155561666677963614442, −2.75454777813083884345021466355, −1.73904721226916918910744268138, −0.53260913879389093446682863135,
0.53260913879389093446682863135, 1.73904721226916918910744268138, 2.75454777813083884345021466355, 3.36991511155561666677963614442, 4.29611386479669596781442295531, 4.89070274351756448540657880811, 5.63723162264792662564034588512, 6.30998251773146079328586458311, 7.34024456024990360313689940413, 7.81046155313820667971289187945
