| L(s) = 1 | − 2·4-s + 5-s + 4·9-s + 4·13-s + 4·16-s − 12·17-s − 2·20-s − 4·25-s − 8·36-s − 2·37-s − 6·41-s + 4·45-s + 2·49-s − 8·52-s − 13·53-s + 10·61-s − 8·64-s + 4·65-s + 24·68-s − 2·73-s + 4·80-s + 7·81-s − 12·85-s + 24·89-s + 10·97-s + 8·100-s − 12·101-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 4/3·9-s + 1.10·13-s + 16-s − 2.91·17-s − 0.447·20-s − 4/5·25-s − 4/3·36-s − 0.328·37-s − 0.937·41-s + 0.596·45-s + 2/7·49-s − 1.10·52-s − 1.78·53-s + 1.28·61-s − 64-s + 0.496·65-s + 2.91·68-s − 0.234·73-s + 0.447·80-s + 7/9·81-s − 1.30·85-s + 2.54·89-s + 1.01·97-s + 4/5·100-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7441133267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7441133267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 12 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.73091995435072935616474322247, −11.87695953386478830377450176328, −11.12861149270607666911621761179, −10.63530562675408550669833007924, −9.979551782259052851675936756200, −9.414247610556808307059138589484, −8.832715740821302498668790162905, −8.389143546910867311269051532984, −7.46243451280049599709785862653, −6.62566816337418312191905909544, −6.14324073284337730711567884401, −4.99155582763338176726655577441, −4.38568537084502361576790861730, −3.68324386515125233033208675776, −1.90044597069967539028129303549,
1.90044597069967539028129303549, 3.68324386515125233033208675776, 4.38568537084502361576790861730, 4.99155582763338176726655577441, 6.14324073284337730711567884401, 6.62566816337418312191905909544, 7.46243451280049599709785862653, 8.389143546910867311269051532984, 8.832715740821302498668790162905, 9.414247610556808307059138589484, 9.979551782259052851675936756200, 10.63530562675408550669833007924, 11.12861149270607666911621761179, 11.87695953386478830377450176328, 12.73091995435072935616474322247
