| L(s) = 1 | + 4-s + 2·7-s + 3·11-s + 16-s − 3·23-s + 2·25-s + 2·28-s + 3·29-s − 11·37-s + 13·43-s + 3·44-s − 3·49-s + 21·53-s + 64-s − 5·67-s + 6·71-s + 6·77-s − 14·79-s − 3·92-s + 2·100-s + 3·107-s + 10·109-s + 2·112-s + 3·116-s − 13·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 0.904·11-s + 1/4·16-s − 0.625·23-s + 2/5·25-s + 0.377·28-s + 0.557·29-s − 1.80·37-s + 1.98·43-s + 0.452·44-s − 3/7·49-s + 2.88·53-s + 1/8·64-s − 0.610·67-s + 0.712·71-s + 0.683·77-s − 1.57·79-s − 0.312·92-s + 1/5·100-s + 0.290·107-s + 0.957·109-s + 0.188·112-s + 0.278·116-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.996840572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.996840572\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 139 T^{2} + p^{2} T^{4} \) |
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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
−8.086957378909260827507725498113, −7.36183531325089434705031299939, −7.12751071510912888473626543776, −6.88293266829974904231241315122, −6.11537814027800075015041152104, −5.91546075633974453763631173072, −5.37028948666120768681653191605, −4.87038393691364837970879749279, −4.30084316977977013470125198039, −3.93637597882520339189729774411, −3.37114565437925398598148669121, −2.69464080839549040218065639632, −2.11200230426453434596424916503, −1.54613197909600252293549042817, −0.799877307541708731708021076720,
0.799877307541708731708021076720, 1.54613197909600252293549042817, 2.11200230426453434596424916503, 2.69464080839549040218065639632, 3.37114565437925398598148669121, 3.93637597882520339189729774411, 4.30084316977977013470125198039, 4.87038393691364837970879749279, 5.37028948666120768681653191605, 5.91546075633974453763631173072, 6.11537814027800075015041152104, 6.88293266829974904231241315122, 7.12751071510912888473626543776, 7.36183531325089434705031299939, 8.086957378909260827507725498113
