L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 12-s − 14-s + 16-s + 6·17-s − 18-s − 8·19-s − 21-s + 24-s − 27-s + 28-s + 6·29-s + 4·31-s − 32-s − 6·34-s + 36-s − 10·37-s + 8·38-s + 6·41-s + 42-s + 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.218·21-s + 0.204·24-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.64·37-s + 1.29·38-s + 0.937·41-s + 0.154·42-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.167576737\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167576737\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.10719228644337, −12.45993967577108, −12.08393478522133, −11.89705635203143, −11.21006474351652, −10.58206420250314, −10.37736465576034, −10.15078282944118, −9.234056493136232, −8.975400795479222, −8.296642112782681, −7.996574959336069, −7.450565282836390, −6.849417774601253, −6.439188819201147, −5.917364909732471, −5.402963984550109, −4.859973434373720, −4.199810619219487, −3.768990683794891, −2.875309194862183, −2.435462628385528, −1.618838866771842, −1.139532079829350, −0.3969761779474491,
0.3969761779474491, 1.139532079829350, 1.618838866771842, 2.435462628385528, 2.875309194862183, 3.768990683794891, 4.199810619219487, 4.859973434373720, 5.402963984550109, 5.917364909732471, 6.439188819201147, 6.849417774601253, 7.450565282836390, 7.996574959336069, 8.296642112782681, 8.975400795479222, 9.234056493136232, 10.15078282944118, 10.37736465576034, 10.58206420250314, 11.21006474351652, 11.89705635203143, 12.08393478522133, 12.45993967577108, 13.10719228644337