| L(s) = 1 | − 3·3-s − 2.82·5-s − 14.1·7-s + 9·9-s + 20·11-s − 39.5·13-s + 8.48·15-s − 34·17-s + 52·19-s + 42.4·21-s − 62.2·23-s − 117·25-s − 27·27-s − 200.·29-s + 110.·31-s − 60·33-s + 40.0·35-s + 271.·37-s + 118.·39-s − 26·41-s + 252·43-s − 25.4·45-s − 345.·47-s − 142.·49-s + 102·51-s + 681.·53-s − 56.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.252·5-s − 0.763·7-s + 0.333·9-s + 0.548·11-s − 0.844·13-s + 0.146·15-s − 0.485·17-s + 0.627·19-s + 0.440·21-s − 0.564·23-s − 0.936·25-s − 0.192·27-s − 1.28·29-s + 0.639·31-s − 0.316·33-s + 0.193·35-s + 1.20·37-s + 0.487·39-s − 0.0990·41-s + 0.893·43-s − 0.0843·45-s − 1.07·47-s − 0.416·49-s + 0.280·51-s + 1.76·53-s − 0.138·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.010299167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010299167\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| good | 5 | \( 1 + 2.82T + 125T^{2} \) |
| 7 | \( 1 + 14.1T + 343T^{2} \) |
| 11 | \( 1 - 20T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 26T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252T + 7.95e4T^{2} \) |
| 47 | \( 1 + 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 681.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 364T + 2.05e5T^{2} \) |
| 61 | \( 1 + 735.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628T + 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 338T + 3.89e5T^{2} \) |
| 79 | \( 1 - 789.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 234T + 7.04e5T^{2} \) |
| 97 | \( 1 + 178T + 9.12e5T^{2} \) |
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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
−9.705020096872218086232039158085, −9.464874960759335128461078658517, −8.085711879257067165617736158214, −7.23860818730725086101304905575, −6.40286643664248732721942221439, −5.57694256957080654905616839450, −4.45071942752871467564195329991, −3.52972691140513077770072419225, −2.16357949390786129934228323707, −0.56654290941226842393521972251,
0.56654290941226842393521972251, 2.16357949390786129934228323707, 3.52972691140513077770072419225, 4.45071942752871467564195329991, 5.57694256957080654905616839450, 6.40286643664248732721942221439, 7.23860818730725086101304905575, 8.085711879257067165617736158214, 9.464874960759335128461078658517, 9.705020096872218086232039158085
