| L(s) = 1 | + 2·3-s + 2·5-s − 3·9-s + 4·15-s + 3·25-s − 14·27-s + 4·31-s − 16·37-s − 6·45-s + 6·47-s − 11·49-s − 24·53-s + 12·59-s − 14·67-s − 12·71-s + 6·75-s − 4·81-s − 18·89-s + 8·93-s − 8·97-s + 32·103-s − 32·111-s − 12·113-s + 4·125-s + 127-s + 131-s − 28·135-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 9-s + 1.03·15-s + 3/5·25-s − 2.69·27-s + 0.718·31-s − 2.63·37-s − 0.894·45-s + 0.875·47-s − 1.57·49-s − 3.29·53-s + 1.56·59-s − 1.71·67-s − 1.42·71-s + 0.692·75-s − 4/9·81-s − 1.90·89-s + 0.829·93-s − 0.812·97-s + 3.15·103-s − 3.03·111-s − 1.12·113-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s − 2.40·135-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−7.53425677247202445915154930283, −7.30153163245751144285916004001, −6.73955568369899846338690720145, −6.54483997956769996864119195578, −6.05756029874302393469627580298, −5.90421539650305055435161749208, −5.39957389208675059275610494019, −5.25930983228565534409246039580, −4.61564492177599448967714947776, −4.56002362830400627900733425879, −3.79852086146731192522019667812, −3.34173877907920924648656678680, −3.26539495356181322856977785868, −2.87653108983217856723446827956, −2.28025381849714230302570465060, −2.22544394005948352741040083431, −1.45890271118594015749355184423, −1.33518402125439306153374988995, 0, 0,
1.33518402125439306153374988995, 1.45890271118594015749355184423, 2.22544394005948352741040083431, 2.28025381849714230302570465060, 2.87653108983217856723446827956, 3.26539495356181322856977785868, 3.34173877907920924648656678680, 3.79852086146731192522019667812, 4.56002362830400627900733425879, 4.61564492177599448967714947776, 5.25930983228565534409246039580, 5.39957389208675059275610494019, 5.90421539650305055435161749208, 6.05756029874302393469627580298, 6.54483997956769996864119195578, 6.73955568369899846338690720145, 7.30153163245751144285916004001, 7.53425677247202445915154930283
