| L(s) = 1 | − 2.41·2-s + 3.82·4-s − 0.828·7-s − 4.41·8-s + 4.82·11-s + 2·13-s + 1.99·14-s + 2.99·16-s − 2.82·17-s + 0.828·19-s − 11.6·22-s − 8.82·23-s − 4.82·26-s − 3.17·28-s − 29-s − 10.4·31-s + 1.58·32-s + 6.82·34-s − 8.48·37-s − 1.99·38-s + 6·41-s + 6·43-s + 18.4·44-s + 21.3·46-s − 0.343·47-s − 6.31·49-s + 7.65·52-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s − 0.313·7-s − 1.56·8-s + 1.45·11-s + 0.554·13-s + 0.534·14-s + 0.749·16-s − 0.685·17-s + 0.190·19-s − 2.48·22-s − 1.84·23-s − 0.946·26-s − 0.599·28-s − 0.185·29-s − 1.88·31-s + 0.280·32-s + 1.17·34-s − 1.39·37-s − 0.324·38-s + 0.937·41-s + 0.914·43-s + 2.78·44-s + 3.14·46-s − 0.0500·47-s − 0.901·49-s + 1.06·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7091451481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7091451481\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
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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
−8.145519897957742037803230067097, −7.46325394231910153867160895862, −6.73170715204644303945319137399, −6.31686255870775567119728404287, −5.46384112041277478470517063406, −4.08398326576206621613594375929, −3.59330364098120732561622493720, −2.22382438145767870959389157923, −1.65682607629540478520366799219, −0.57104723809250590467170896966,
0.57104723809250590467170896966, 1.65682607629540478520366799219, 2.22382438145767870959389157923, 3.59330364098120732561622493720, 4.08398326576206621613594375929, 5.46384112041277478470517063406, 6.31686255870775567119728404287, 6.73170715204644303945319137399, 7.46325394231910153867160895862, 8.145519897957742037803230067097
