| L(s) = 1 | − 343·7-s + 729·9-s − 1.96e3·11-s − 2.27e4·23-s + 1.56e4·25-s + 2.12e4·29-s − 1.01e5·37-s + 1.26e5·43-s + 1.17e5·49-s − 5.03e4·53-s − 2.50e5·63-s + 5.39e4·67-s − 2.42e5·71-s + 6.72e5·77-s + 9.29e5·79-s + 5.31e5·81-s − 1.43e6·99-s − 4.63e4·107-s + 2.58e6·109-s − 2.43e6·113-s + ⋯ |
| L(s) = 1 | − 7-s + 9-s − 1.47·11-s − 1.86·23-s + 25-s + 0.870·29-s − 1.99·37-s + 1.59·43-s + 49-s − 0.338·53-s − 63-s + 0.179·67-s − 0.677·71-s + 1.47·77-s + 1.88·79-s + 81-s − 1.47·99-s − 0.0378·107-s + 1.99·109-s − 1.68·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.382448886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382448886\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( 1 + 1962 T + p^{6} T^{2} \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( 1 + 22734 T + p^{6} T^{2} \) |
| 29 | \( 1 - 21222 T + p^{6} T^{2} \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( 1 + 101194 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 - 126614 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( 1 + 50346 T + p^{6} T^{2} \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( 1 - 53926 T + p^{6} T^{2} \) |
| 71 | \( 1 + 242478 T + p^{6} T^{2} \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( 1 - 929378 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.29811410586057071139425622368, −9.348690004421762255697504218819, −8.222354139092689793901129121379, −7.33244386776425823432589364066, −6.44991133211699588807658475222, −5.39416999671670282983267721264, −4.29063100298125649108803899415, −3.16491777356837267367177442680, −2.06889446689533713327536368082, −0.54302600204891214034835210131,
0.54302600204891214034835210131, 2.06889446689533713327536368082, 3.16491777356837267367177442680, 4.29063100298125649108803899415, 5.39416999671670282983267721264, 6.44991133211699588807658475222, 7.33244386776425823432589364066, 8.222354139092689793901129121379, 9.348690004421762255697504218819, 10.29811410586057071139425622368
