L(s) = 1 | + 3-s − 2·5-s + 9-s − 11-s + 2·13-s − 2·15-s − 4·17-s − 6·19-s − 25-s + 27-s − 8·29-s − 8·31-s − 33-s + 10·37-s + 2·39-s − 8·41-s + 2·43-s − 2·45-s − 8·47-s − 4·51-s − 2·53-s + 2·55-s − 6·57-s + 12·59-s − 10·61-s − 4·65-s − 12·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.516·15-s − 0.970·17-s − 1.37·19-s − 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s + 1.64·37-s + 0.320·39-s − 1.24·41-s + 0.304·43-s − 0.298·45-s − 1.16·47-s − 0.560·51-s − 0.274·53-s + 0.269·55-s − 0.794·57-s + 1.56·59-s − 1.28·61-s − 0.496·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.003998050\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003998050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.16326380357521, −14.87813915775571, −14.52803511436998, −13.50964903361713, −13.09268940134022, −12.97369739106917, −12.02644188493257, −11.53456166660604, −10.93205166626398, −10.63706444113735, −9.793130170889716, −9.113184373411291, −8.745550606297384, −8.122087642021888, −7.617859731737542, −7.129917196125340, −6.329280283777802, −5.848904660662853, −4.844101429850706, −4.316210959903267, −3.727197067048951, −3.211475296055988, −2.195440365873814, −1.733427515058483, −0.3624833271469523,
0.3624833271469523, 1.733427515058483, 2.195440365873814, 3.211475296055988, 3.727197067048951, 4.316210959903267, 4.844101429850706, 5.848904660662853, 6.329280283777802, 7.129917196125340, 7.617859731737542, 8.122087642021888, 8.745550606297384, 9.113184373411291, 9.793130170889716, 10.63706444113735, 10.93205166626398, 11.53456166660604, 12.02644188493257, 12.97369739106917, 13.09268940134022, 13.50964903361713, 14.52803511436998, 14.87813915775571, 15.16326380357521