| L(s) = 1 | + 3-s + 5-s − 5·7-s + 9-s − 13-s + 15-s − 6·17-s + 5·19-s − 5·21-s − 4·23-s + 25-s + 27-s + 6·29-s + 9·31-s − 5·35-s + 6·37-s − 39-s − 2·41-s − 11·43-s + 45-s + 3·47-s + 18·49-s − 6·51-s + 5·57-s + 5·59-s − 3·61-s − 5·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.88·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 1.45·17-s + 1.14·19-s − 1.09·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.61·31-s − 0.845·35-s + 0.986·37-s − 0.160·39-s − 0.312·41-s − 1.67·43-s + 0.149·45-s + 0.437·47-s + 18/7·49-s − 0.840·51-s + 0.662·57-s + 0.650·59-s − 0.384·61-s − 0.629·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028748750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028748750\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.17676523498370, −12.85347245917806, −12.16314976640867, −11.86905480276511, −11.33316100084999, −10.41467800469470, −10.18477118983664, −9.856214491966905, −9.273525230853881, −9.071749322865990, −8.345364341750305, −7.978679829242983, −7.196399887053138, −6.733588618674026, −6.458659096074300, −5.989032694382026, −5.335353833795763, −4.587676413816785, −4.227754998508867, −3.436930618081401, −3.040503571890818, −2.569328735850861, −2.067320268372382, −1.113687684559723, −0.4128168410658177,
0.4128168410658177, 1.113687684559723, 2.067320268372382, 2.569328735850861, 3.040503571890818, 3.436930618081401, 4.227754998508867, 4.587676413816785, 5.335353833795763, 5.989032694382026, 6.458659096074300, 6.733588618674026, 7.196399887053138, 7.978679829242983, 8.345364341750305, 9.071749322865990, 9.273525230853881, 9.856214491966905, 10.18477118983664, 10.41467800469470, 11.33316100084999, 11.86905480276511, 12.16314976640867, 12.85347245917806, 13.17676523498370
