| L(s) = 1 | + 16·4-s + 192·16-s + 326·19-s − 38·31-s − 683·49-s + 1.43e3·61-s + 2.04e3·64-s + 5.21e3·76-s − 1.00e3·79-s − 4.42e3·109-s − 2.66e3·121-s − 608·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s + 3.93·19-s − 0.220·31-s − 1.99·49-s + 3.01·61-s + 4·64-s + 7.87·76-s − 1.43·79-s − 3.88·109-s − 2·121-s − 0.440·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.230·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.295445034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.295445034\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 683 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 163 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 86183 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 42587 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 719 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 172874 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 704593 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 503 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1551817 T^{2} + p^{6} T^{4} \) |
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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
−10.17086911743834983054082781157, −10.07632922561253161996693268573, −9.494452201942319938462188342780, −9.289817213313369317696058893857, −8.355118521535770959554939520664, −7.84586098052872572990272910799, −7.77881913353151303155891928591, −7.15096164573269241705698939406, −6.79041699374872039093559715349, −6.58555901472753891567990605024, −5.65050733363317889129087755091, −5.46370834143739752590778046173, −5.22771646484566273395918750131, −4.16731921881620373603199531963, −3.41007351263224283719934570235, −3.11546848500810103180832529627, −2.71628928804755121360437384943, −1.87554446615192534822022068435, −1.31892464296328645811167867345, −0.800101163494392080922744910495,
0.800101163494392080922744910495, 1.31892464296328645811167867345, 1.87554446615192534822022068435, 2.71628928804755121360437384943, 3.11546848500810103180832529627, 3.41007351263224283719934570235, 4.16731921881620373603199531963, 5.22771646484566273395918750131, 5.46370834143739752590778046173, 5.65050733363317889129087755091, 6.58555901472753891567990605024, 6.79041699374872039093559715349, 7.15096164573269241705698939406, 7.77881913353151303155891928591, 7.84586098052872572990272910799, 8.355118521535770959554939520664, 9.289817213313369317696058893857, 9.494452201942319938462188342780, 10.07632922561253161996693268573, 10.17086911743834983054082781157
