| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s + 23-s − 24-s + 25-s − 27-s − 29-s − 30-s + 31-s + 32-s + 34-s + 36-s − 37-s − 38-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s + 23-s − 24-s + 25-s − 27-s − 29-s − 30-s + 31-s + 32-s + 34-s + 36-s − 37-s − 38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.769093905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.769093905\) |
| \(L(1)\) |
\(\approx\) |
\(1.841773960\) |
| \(L(1)\) |
\(\approx\) |
\(1.841773960\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.74265820940144991375610658330, −21.07973176848511201735198743885, −20.61231707892567028872444756086, −19.19138160772353630767465267719, −18.59458658508285972006357527853, −17.33777579941455901617602829777, −16.9968444981433234078000112948, −16.235352219805653398217336415453, −15.202821695454490413486338059747, −14.56071080478199112716861071539, −13.46071230264861468027506877627, −13.016851790964565505291688841680, −12.15021519817850063502862935162, −11.43215835184530768689945169027, −10.40055900696255424945177913962, −10.092240638085773541022174031013, −8.701009115658206880488133545666, −7.2945821673295489930636741938, −6.65564409224680009661348867643, −5.78653453316782703632001445914, −5.27498182371009353634645780847, −4.401062626074685193700047697490, −3.27628639458399689037544550932, −2.09408684144974849212636822918, −1.19827459870646569310156473869,
1.19827459870646569310156473869, 2.09408684144974849212636822918, 3.27628639458399689037544550932, 4.401062626074685193700047697490, 5.27498182371009353634645780847, 5.78653453316782703632001445914, 6.65564409224680009661348867643, 7.2945821673295489930636741938, 8.701009115658206880488133545666, 10.092240638085773541022174031013, 10.40055900696255424945177913962, 11.43215835184530768689945169027, 12.15021519817850063502862935162, 13.016851790964565505291688841680, 13.46071230264861468027506877627, 14.56071080478199112716861071539, 15.202821695454490413486338059747, 16.235352219805653398217336415453, 16.9968444981433234078000112948, 17.33777579941455901617602829777, 18.59458658508285972006357527853, 19.19138160772353630767465267719, 20.61231707892567028872444756086, 21.07973176848511201735198743885, 21.74265820940144991375610658330
