L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s − 4·11-s − 13-s + 15-s + 4·17-s + 2·19-s + 2·21-s − 2·23-s + 25-s + 27-s + 8·29-s − 4·31-s − 4·33-s + 2·35-s + 6·37-s − 39-s + 10·41-s − 4·43-s + 45-s − 3·49-s + 4·51-s + 6·53-s − 4·55-s + 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.258·15-s + 0.970·17-s + 0.458·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.718·31-s − 0.696·33-s + 0.338·35-s + 0.986·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 3/7·49-s + 0.560·51-s + 0.824·53-s − 0.539·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.735865669\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.735865669\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.549049893125375715459393748304, −7.910154235688354030561524918975, −7.50153755168620563027857335894, −6.44527682854287992609356702402, −5.45249746742298175225580370510, −4.97799698305641515989194911832, −3.95540701188570399836856987203, −2.86632359514013041022785305060, −2.21848090724422075370338433518, −1.01699316534771071213032644023,
1.01699316534771071213032644023, 2.21848090724422075370338433518, 2.86632359514013041022785305060, 3.95540701188570399836856987203, 4.97799698305641515989194911832, 5.45249746742298175225580370510, 6.44527682854287992609356702402, 7.50153755168620563027857335894, 7.910154235688354030561524918975, 8.549049893125375715459393748304