L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s − 3·7-s − 3·8-s − 2·9-s − 2·10-s + 11-s − 12-s + 13-s − 3·14-s − 2·15-s − 16-s − 2·18-s − 4·19-s + 2·20-s − 3·21-s + 22-s − 3·23-s − 3·24-s − 25-s + 26-s − 5·27-s + 3·28-s − 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s − 1.06·8-s − 2/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.516·15-s − 1/4·16-s − 0.471·18-s − 0.917·19-s + 0.447·20-s − 0.654·21-s + 0.213·22-s − 0.625·23-s − 0.612·24-s − 1/5·25-s + 0.196·26-s − 0.962·27-s + 0.566·28-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 503 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.42865847180454753626482692579, −9.425911885927882691529707835789, −8.665748270161512723514477907441, −7.969191094446663725578986119675, −6.58017975079940646660645615550, −5.80994036929178904110160722019, −4.38683468515013665070855308133, −3.64649038386658239035532348038, −2.79850124252583110146349421753, 0,
2.79850124252583110146349421753, 3.64649038386658239035532348038, 4.38683468515013665070855308133, 5.80994036929178904110160722019, 6.58017975079940646660645615550, 7.969191094446663725578986119675, 8.665748270161512723514477907441, 9.425911885927882691529707835789, 10.42865847180454753626482692579