| L(s) = 1 | + 2-s − 4·7-s − 8-s + 3·11-s − 4·13-s − 4·14-s − 16-s − 6·17-s − 8·19-s + 3·22-s + 6·23-s − 4·26-s − 6·29-s − 8·31-s − 6·34-s − 16·37-s − 8·38-s − 6·41-s − 43-s + 6·46-s − 12·47-s + 7·49-s + 4·56-s − 6·58-s − 9·59-s − 8·61-s − 8·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.51·7-s − 0.353·8-s + 0.904·11-s − 1.10·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s − 1.83·19-s + 0.639·22-s + 1.25·23-s − 0.784·26-s − 1.11·29-s − 1.43·31-s − 1.02·34-s − 2.63·37-s − 1.29·38-s − 0.937·41-s − 0.152·43-s + 0.884·46-s − 1.75·47-s + 49-s + 0.534·56-s − 0.787·58-s − 1.17·59-s − 1.02·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.209949135907358853707789453404, −9.072565705672094122840826390806, −8.716156967957626730750474271484, −8.492830839370149759457323954936, −7.47226948155862627954468333886, −7.21479440082634321312809627087, −6.92379470558340925961329827373, −6.33940320284483676222591412271, −6.30019813578637055781990469776, −5.72249825854272382114369910028, −4.94303188473312959133888086182, −4.81967146148831469765363247224, −4.30963274915364172806476699329, −3.59923591723418374851085855663, −3.41453047061195233080945793959, −2.93649478361689139650692007268, −1.94679271358269380536154130089, −1.86973378363312815574764383229, 0, 0,
1.86973378363312815574764383229, 1.94679271358269380536154130089, 2.93649478361689139650692007268, 3.41453047061195233080945793959, 3.59923591723418374851085855663, 4.30963274915364172806476699329, 4.81967146148831469765363247224, 4.94303188473312959133888086182, 5.72249825854272382114369910028, 6.30019813578637055781990469776, 6.33940320284483676222591412271, 6.92379470558340925961329827373, 7.21479440082634321312809627087, 7.47226948155862627954468333886, 8.492830839370149759457323954936, 8.716156967957626730750474271484, 9.072565705672094122840826390806, 9.209949135907358853707789453404
