| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 9-s + 14-s + 16-s − 3·17-s + 18-s − 15·23-s − 2·25-s − 28-s − 12·31-s − 32-s + 3·34-s − 36-s − 3·41-s + 15·46-s + 9·47-s + 49-s + 2·50-s + 56-s + 12·62-s + 63-s + 64-s − 3·68-s + 9·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1/3·9-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 3.12·23-s − 2/5·25-s − 0.188·28-s − 2.15·31-s − 0.176·32-s + 0.514·34-s − 1/6·36-s − 0.468·41-s + 2.21·46-s + 1.31·47-s + 1/7·49-s + 0.282·50-s + 0.133·56-s + 1.52·62-s + 0.125·63-s + 1/8·64-s − 0.363·68-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−9.903259475993730301133601247932, −9.400946509192316215215642760736, −9.032872625967611552960835581299, −8.322439564034960678023719361421, −7.986082184770432412320687121706, −7.40607211260737083787304764859, −6.80999099230277597284059767890, −6.15879308148576935335469093848, −5.78902777848912703504487358241, −5.08639353379536435232603331513, −3.92316526135323766397833934784, −3.75383713546245393225954256150, −2.46401242958316858193289692726, −1.88856445010374489988253513893, 0,
1.88856445010374489988253513893, 2.46401242958316858193289692726, 3.75383713546245393225954256150, 3.92316526135323766397833934784, 5.08639353379536435232603331513, 5.78902777848912703504487358241, 6.15879308148576935335469093848, 6.80999099230277597284059767890, 7.40607211260737083787304764859, 7.986082184770432412320687121706, 8.322439564034960678023719361421, 9.032872625967611552960835581299, 9.400946509192316215215642760736, 9.903259475993730301133601247932
