| L(s) = 1 | + 1.67·2-s + 0.795·4-s − 5.24·7-s − 2.01·8-s − 3.01·11-s − 3.80·13-s − 8.76·14-s − 4.95·16-s − 4.30·17-s + 2.28·19-s − 5.04·22-s − 2.78·23-s − 6.36·26-s − 4.17·28-s + 1.29·29-s + 6.11·31-s − 4.26·32-s − 7.19·34-s − 37-s + 3.81·38-s − 1.76·41-s − 0.682·43-s − 2.40·44-s − 4.64·46-s + 11.3·47-s + 20.4·49-s − 3.02·52-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.397·4-s − 1.98·7-s − 0.711·8-s − 0.909·11-s − 1.05·13-s − 2.34·14-s − 1.23·16-s − 1.04·17-s + 0.523·19-s − 1.07·22-s − 0.579·23-s − 1.24·26-s − 0.788·28-s + 0.240·29-s + 1.09·31-s − 0.753·32-s − 1.23·34-s − 0.164·37-s + 0.619·38-s − 0.275·41-s − 0.104·43-s − 0.362·44-s − 0.685·46-s + 1.65·47-s + 2.92·49-s − 0.419·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8900048495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8900048495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 7 | \( 1 + 5.24T + 7T^{2} \) |
| 11 | \( 1 + 3.01T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 41 | \( 1 + 1.76T + 41T^{2} \) |
| 43 | \( 1 + 0.682T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 5.45T + 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 5.51T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 + 8.61T + 83T^{2} \) |
| 89 | \( 1 + 9.28T + 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−7.46171467756760871189723065730, −6.95585642197847856044005308270, −6.13982114935591429708911524421, −5.85108429098989901941979175855, −4.86767081444543125014506897101, −4.36759256176708084382441769219, −3.44537510967848644640099365491, −2.86256887997655661548037579561, −2.35024389299189992409962044453, −0.35181810483633992261814687811,
0.35181810483633992261814687811, 2.35024389299189992409962044453, 2.86256887997655661548037579561, 3.44537510967848644640099365491, 4.36759256176708084382441769219, 4.86767081444543125014506897101, 5.85108429098989901941979175855, 6.13982114935591429708911524421, 6.95585642197847856044005308270, 7.46171467756760871189723065730
