L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 3·7-s − 3·8-s − 2·9-s + 10-s + 4·11-s − 12-s + 3·14-s + 15-s − 16-s + 17-s − 2·18-s + 7·19-s − 20-s + 3·21-s + 4·22-s + 2·23-s − 3·24-s − 4·25-s − 5·27-s − 3·28-s + 3·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.801·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.471·18-s + 1.60·19-s − 0.223·20-s + 0.654·21-s + 0.852·22-s + 0.417·23-s − 0.612·24-s − 4/5·25-s − 0.962·27-s − 0.566·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.831526143\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.831526143\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.620361622323635875842436276164, −9.211483002218959705833606295063, −8.368466260100878773851317210974, −7.62918809679017411561616119417, −6.30171393316144630625973682715, −5.49867939231750205477734291705, −4.71804579054670114481457460818, −3.72221808916004673981279320783, −2.79566817168447846102691997335, −1.33758264545779565551971807318,
1.33758264545779565551971807318, 2.79566817168447846102691997335, 3.72221808916004673981279320783, 4.71804579054670114481457460818, 5.49867939231750205477734291705, 6.30171393316144630625973682715, 7.62918809679017411561616119417, 8.368466260100878773851317210974, 9.211483002218959705833606295063, 9.620361622323635875842436276164