| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 13-s − 15-s + 2·17-s − 2·21-s − 7·23-s + 25-s + 27-s + 5·31-s + 2·35-s − 8·37-s + 39-s − 6·41-s + 10·43-s − 45-s − 10·47-s − 3·49-s + 2·51-s − 9·53-s − 4·59-s + 10·61-s − 2·63-s − 65-s + 67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.485·17-s − 0.436·21-s − 1.45·23-s + 1/5·25-s + 0.192·27-s + 0.898·31-s + 0.338·35-s − 1.31·37-s + 0.160·39-s − 0.937·41-s + 1.52·43-s − 0.149·45-s − 1.45·47-s − 3/7·49-s + 0.280·51-s − 1.23·53-s − 0.520·59-s + 1.28·61-s − 0.251·63-s − 0.124·65-s + 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.77785582709168, −12.24252946000482, −11.95070725990679, −11.32968795913677, −10.98095814368390, −10.18784655728057, −10.01960846578504, −9.672508442081725, −9.003133911122716, −8.587430339914372, −8.163361940164385, −7.742106367781753, −7.309341405036992, −6.610409943141936, −6.395358459585594, −5.829549554664831, −5.176797267142744, −4.689815344035795, −4.046723803749392, −3.656484855191137, −3.189054995586554, −2.758797283222481, −1.979128416799337, −1.534535399328030, −0.6770447684682869, 0,
0.6770447684682869, 1.534535399328030, 1.979128416799337, 2.758797283222481, 3.189054995586554, 3.656484855191137, 4.046723803749392, 4.689815344035795, 5.176797267142744, 5.829549554664831, 6.395358459585594, 6.610409943141936, 7.309341405036992, 7.742106367781753, 8.163361940164385, 8.587430339914372, 9.003133911122716, 9.672508442081725, 10.01960846578504, 10.18784655728057, 10.98095814368390, 11.32968795913677, 11.95070725990679, 12.24252946000482, 12.77785582709168
