| L(s) = 1 | + 2·2-s + 4·4-s + 22.2·7-s + 8·8-s + 22.2·11-s − 66.8·13-s + 44.5·14-s + 16·16-s + 62·17-s + 84·19-s + 44.5·22-s + 140·23-s − 133.·26-s + 89.0·28-s − 200.·29-s + 16·31-s + 32·32-s + 124·34-s − 244.·37-s + 168·38-s + 222.·41-s + 356.·43-s + 89.0·44-s + 280·46-s + 100·47-s + 152.·49-s − 267.·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.20·7-s + 0.353·8-s + 0.610·11-s − 1.42·13-s + 0.850·14-s + 0.250·16-s + 0.884·17-s + 1.01·19-s + 0.431·22-s + 1.26·23-s − 1.00·26-s + 0.601·28-s − 1.28·29-s + 0.0926·31-s + 0.176·32-s + 0.625·34-s − 1.08·37-s + 0.717·38-s + 0.848·41-s + 1.26·43-s + 0.305·44-s + 0.897·46-s + 0.310·47-s + 0.446·49-s − 0.712·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.627453624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.627453624\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 22.2T + 343T^{2} \) |
| 11 | \( 1 - 22.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 16T + 2.97e4T^{2} \) |
| 37 | \( 1 + 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 100T + 1.03e5T^{2} \) |
| 53 | \( 1 - 738T + 1.48e5T^{2} \) |
| 59 | \( 1 - 645.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 358T + 2.26e5T^{2} \) |
| 67 | \( 1 - 846.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 935.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 445.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 936T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 712.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 757.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91325510212989642997366542315, −9.855758807957044587102758869553, −8.877068859602440315619710431198, −7.57343723924811754597840143077, −7.16902440058342457029628776373, −5.58037578033031578540529797356, −5.01034348210061453900338334788, −3.88596975781471889340555873069, −2.55143500753069135015772712354, −1.22000691428943146410391995160,
1.22000691428943146410391995160, 2.55143500753069135015772712354, 3.88596975781471889340555873069, 5.01034348210061453900338334788, 5.58037578033031578540529797356, 7.16902440058342457029628776373, 7.57343723924811754597840143077, 8.877068859602440315619710431198, 9.855758807957044587102758869553, 10.91325510212989642997366542315
