| L(s) = 1 | − 2.10·3-s + 3.40·5-s + 7-s + 1.43·9-s − 1.23·11-s + 6.47·13-s − 7.17·15-s + 17-s − 2.21·19-s − 2.10·21-s − 1.39·23-s + 6.60·25-s + 3.30·27-s + 0.633·29-s − 0.965·31-s + 2.60·33-s + 3.40·35-s + 8.31·37-s − 13.6·39-s − 5.60·41-s + 11.0·43-s + 4.87·45-s − 7.67·47-s + 49-s − 2.10·51-s − 11.7·53-s − 4.21·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s + 1.52·5-s + 0.377·7-s + 0.477·9-s − 0.372·11-s + 1.79·13-s − 1.85·15-s + 0.242·17-s − 0.507·19-s − 0.459·21-s − 0.291·23-s + 1.32·25-s + 0.635·27-s + 0.117·29-s − 0.173·31-s + 0.452·33-s + 0.575·35-s + 1.36·37-s − 2.18·39-s − 0.875·41-s + 1.68·43-s + 0.727·45-s − 1.11·47-s + 0.142·49-s − 0.294·51-s − 1.60·53-s − 0.567·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.040613599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040613599\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.10T + 3T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 - 0.633T + 29T^{2} \) |
| 31 | \( 1 + 0.965T + 31T^{2} \) |
| 37 | \( 1 - 8.31T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 0.602T + 59T^{2} \) |
| 61 | \( 1 - 9.84T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 - 3.33T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 0.323T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 5.90T + 97T^{2} \) |
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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
−7.984474102715206700467574394777, −6.78507741276266580927668317707, −6.24682695793003267178746157198, −5.87948261803524795544584972602, −5.29555486191029549863807084369, −4.60256829739869630211230909109, −3.59337191285530767165271811397, −2.49044285825343480039583758197, −1.61064578947545683983213250751, −0.811327393420747048358085619522,
0.811327393420747048358085619522, 1.61064578947545683983213250751, 2.49044285825343480039583758197, 3.59337191285530767165271811397, 4.60256829739869630211230909109, 5.29555486191029549863807084369, 5.87948261803524795544584972602, 6.24682695793003267178746157198, 6.78507741276266580927668317707, 7.984474102715206700467574394777
