| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 14-s + 16-s − 9·17-s − 3·23-s − 4·25-s + 28-s − 5·31-s − 32-s + 9·34-s + 6·41-s + 3·46-s − 15·47-s − 11·49-s + 4·50-s − 56-s + 5·62-s + 64-s − 9·68-s − 9·71-s − 2·73-s + 7·79-s − 6·82-s + 18·89-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.267·14-s + 1/4·16-s − 2.18·17-s − 0.625·23-s − 4/5·25-s + 0.188·28-s − 0.898·31-s − 0.176·32-s + 1.54·34-s + 0.937·41-s + 0.442·46-s − 2.18·47-s − 1.57·49-s + 0.565·50-s − 0.133·56-s + 0.635·62-s + 1/8·64-s − 1.09·68-s − 1.06·71-s − 0.234·73-s + 0.787·79-s − 0.662·82-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.234784731291582893942336185722, −9.047619493263235558778481775285, −8.299898036620147952797155134652, −8.028008160289028286732799087973, −7.49292253877125926898454695228, −6.83120361300625922565171947524, −6.42976060527857294030104406094, −5.95127792046835865099983631792, −5.14999272117293209476249264247, −4.57258363417676893879697044435, −3.98028122627839318724648505063, −3.15726402555359122983802556008, −2.21712650517088164673241113134, −1.70081546976403490823648899571, 0,
1.70081546976403490823648899571, 2.21712650517088164673241113134, 3.15726402555359122983802556008, 3.98028122627839318724648505063, 4.57258363417676893879697044435, 5.14999272117293209476249264247, 5.95127792046835865099983631792, 6.42976060527857294030104406094, 6.83120361300625922565171947524, 7.49292253877125926898454695228, 8.028008160289028286732799087973, 8.299898036620147952797155134652, 9.047619493263235558778481775285, 9.234784731291582893942336185722
