L(s) = 1 | − 8·5-s − 8·23-s + 38·25-s − 6·31-s + 2·37-s + 8·41-s − 18·43-s − 2·49-s − 16·59-s + 14·61-s − 26·73-s − 24·83-s + 22·103-s − 8·113-s + 64·115-s + 13·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·155-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 3.57·5-s − 1.66·23-s + 38/5·25-s − 1.07·31-s + 0.328·37-s + 1.24·41-s − 2.74·43-s − 2/7·49-s − 2.08·59-s + 1.79·61-s − 3.04·73-s − 2.63·83-s + 2.16·103-s − 0.752·113-s + 5.96·115-s + 1.18·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.85·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 41 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 141 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−9.102057242194324532889608306317, −8.583186023153649165460373595080, −8.449094410328137953880527944836, −8.118777053742042285317924728597, −7.59073884818765709926116770914, −7.53160673080902664424401454310, −6.98490287126549412715468122986, −6.79398682899539228030846400398, −5.95394242748069983084251539736, −5.64823513073751212840566002703, −4.73907683203477975564294956292, −4.49003882994578320873290918364, −4.28259567976784924609768445875, −3.66761260215031205692779651330, −3.33024698704583651930601066548, −3.06824639117645043782410067366, −2.07191517393699550808111049291, −1.13348383262806300238633910171, 0, 0,
1.13348383262806300238633910171, 2.07191517393699550808111049291, 3.06824639117645043782410067366, 3.33024698704583651930601066548, 3.66761260215031205692779651330, 4.28259567976784924609768445875, 4.49003882994578320873290918364, 4.73907683203477975564294956292, 5.64823513073751212840566002703, 5.95394242748069983084251539736, 6.79398682899539228030846400398, 6.98490287126549412715468122986, 7.53160673080902664424401454310, 7.59073884818765709926116770914, 8.118777053742042285317924728597, 8.449094410328137953880527944836, 8.583186023153649165460373595080, 9.102057242194324532889608306317