| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 4·9-s + 5·16-s − 8·18-s + 12·23-s + 12·29-s + 6·32-s − 12·36-s + 12·37-s + 24·43-s + 24·46-s − 12·53-s + 24·58-s + 7·64-s + 12·71-s − 16·72-s + 24·74-s − 20·79-s + 7·81-s + 48·86-s + 36·92-s − 24·106-s − 4·109-s + 36·113-s + 36·116-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 4/3·9-s + 5/4·16-s − 1.88·18-s + 2.50·23-s + 2.22·29-s + 1.06·32-s − 2·36-s + 1.97·37-s + 3.65·43-s + 3.53·46-s − 1.64·53-s + 3.15·58-s + 7/8·64-s + 1.42·71-s − 1.88·72-s + 2.78·74-s − 2.25·79-s + 7/9·81-s + 5.17·86-s + 3.75·92-s − 2.33·106-s − 0.383·109-s + 3.38·113-s + 3.34·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.438635238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.438635238\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−9.019670505779790672816517838744, −8.766798130805968109046704672480, −8.312898819076376913363095759417, −7.925379766150347932541290058236, −7.37271998442593291735691271897, −7.28339236859981557561454961886, −6.53312344544757060796474865140, −6.40888460698946926827754316453, −5.86768002058265817649048487513, −5.71943854513071216602311788206, −5.07021925012516361208978147584, −4.83937694972565554770101594754, −4.36232685110883718481838660303, −4.07497151286776616042894507502, −3.18954180703198252545995699763, −3.07642434613909907597530519817, −2.57143776317776818554336913010, −2.36434866636592381548694087464, −1.18784989357059655712253803189, −0.816655776271033166365496871235,
0.816655776271033166365496871235, 1.18784989357059655712253803189, 2.36434866636592381548694087464, 2.57143776317776818554336913010, 3.07642434613909907597530519817, 3.18954180703198252545995699763, 4.07497151286776616042894507502, 4.36232685110883718481838660303, 4.83937694972565554770101594754, 5.07021925012516361208978147584, 5.71943854513071216602311788206, 5.86768002058265817649048487513, 6.40888460698946926827754316453, 6.53312344544757060796474865140, 7.28339236859981557561454961886, 7.37271998442593291735691271897, 7.925379766150347932541290058236, 8.312898819076376913363095759417, 8.766798130805968109046704672480, 9.019670505779790672816517838744
