| L(s) = 1 | − 3-s + 5-s + 3·7-s + 9-s + 3·11-s − 13-s − 15-s − 17-s + 6·19-s − 3·21-s + 5·23-s + 25-s − 27-s − 6·29-s − 2·31-s − 3·33-s + 3·35-s + 7·37-s + 39-s + 3·41-s + 8·43-s + 45-s + 2·47-s + 2·49-s + 51-s − 53-s + 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.258·15-s − 0.242·17-s + 1.37·19-s − 0.654·21-s + 1.04·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.522·33-s + 0.507·35-s + 1.15·37-s + 0.160·39-s + 0.468·41-s + 1.21·43-s + 0.149·45-s + 0.291·47-s + 2/7·49-s + 0.140·51-s − 0.137·53-s + 0.404·55-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
| \(L(1)\) |
\(\approx\) |
\(2.162578387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162578387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−8.990621268354504664137249974914, −7.59847037530222701323007470524, −7.42332238365859225321922402432, −6.28687226943988885056397444100, −5.64126731444850928762435779883, −4.88056016486234565880334549820, −4.22204460101415952340957629738, −3.03918501221305522922865651854, −1.81018363181872635809102388337, −1.00237713841400243429623643867,
1.00237713841400243429623643867, 1.81018363181872635809102388337, 3.03918501221305522922865651854, 4.22204460101415952340957629738, 4.88056016486234565880334549820, 5.64126731444850928762435779883, 6.28687226943988885056397444100, 7.42332238365859225321922402432, 7.59847037530222701323007470524, 8.990621268354504664137249974914
