| L(s) = 1 | − 2·5-s + 7-s − 3·9-s + 11-s − 13-s + 4·17-s − 4·23-s − 25-s − 2·35-s + 4·37-s − 2·41-s + 6·45-s + 4·47-s + 49-s − 2·53-s − 2·55-s − 6·59-s − 2·61-s − 3·63-s + 2·65-s − 4·67-s − 6·71-s − 10·73-s + 77-s + 10·79-s + 9·81-s − 8·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s + 0.301·11-s − 0.277·13-s + 0.970·17-s − 0.834·23-s − 1/5·25-s − 0.338·35-s + 0.657·37-s − 0.312·41-s + 0.894·45-s + 0.583·47-s + 1/7·49-s − 0.274·53-s − 0.269·55-s − 0.781·59-s − 0.256·61-s − 0.377·63-s + 0.248·65-s − 0.488·67-s − 0.712·71-s − 1.17·73-s + 0.113·77-s + 1.12·79-s + 81-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.213121702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.213121702\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.65493527621517633200436975036, −7.53488411810317593825974918828, −6.32365974830909485072899313624, −5.81587277052130304369081586210, −5.00136440050701025611972415586, −4.24216315003717477915223276802, −3.52049696458260007183847764191, −2.80590917498411609221487219981, −1.75972647542623907939996038399, −0.53839986360011735236518588363,
0.53839986360011735236518588363, 1.75972647542623907939996038399, 2.80590917498411609221487219981, 3.52049696458260007183847764191, 4.24216315003717477915223276802, 5.00136440050701025611972415586, 5.81587277052130304369081586210, 6.32365974830909485072899313624, 7.53488411810317593825974918828, 7.65493527621517633200436975036
