| L(s) = 1 | − 2-s + 3-s − 4-s − 4·5-s − 6-s − 3·7-s + 3·8-s − 2·9-s + 4·10-s + 5·11-s − 12-s + 13-s + 3·14-s − 4·15-s − 16-s + 2·18-s + 8·19-s + 4·20-s − 3·21-s − 5·22-s + 9·23-s + 3·24-s + 11·25-s − 26-s − 5·27-s + 3·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.78·5-s − 0.408·6-s − 1.13·7-s + 1.06·8-s − 2/3·9-s + 1.26·10-s + 1.50·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s − 1.03·15-s − 1/4·16-s + 0.471·18-s + 1.83·19-s + 0.894·20-s − 0.654·21-s − 1.06·22-s + 1.87·23-s + 0.612·24-s + 11/5·25-s − 0.196·26-s − 0.962·27-s + 0.566·28-s − 1.11·29-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
| \(L(1)\) |
\(\approx\) |
\(0.6270525695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6270525695\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.06206081241687198712059785558, −9.519039570832559235325197740386, −9.189595554204272005736304162731, −8.374675267475255348467143401000, −7.50965148920699347602649631467, −6.77506720034845754190428900858, −5.09210320628727875983058623774, −3.61986319825680675008831346732, −3.44495927695587183079579052738, −0.789855560627387978746855063115,
0.789855560627387978746855063115, 3.44495927695587183079579052738, 3.61986319825680675008831346732, 5.09210320628727875983058623774, 6.77506720034845754190428900858, 7.50965148920699347602649631467, 8.374675267475255348467143401000, 9.189595554204272005736304162731, 9.519039570832559235325197740386, 11.06206081241687198712059785558
