| L(s) = 1 | − 2·5-s + 4·7-s − 2·13-s − 2·17-s + 8·23-s − 25-s + 6·29-s + 8·31-s − 8·35-s − 6·37-s − 2·41-s + 8·47-s + 9·49-s + 6·53-s + 4·59-s + 6·61-s + 4·65-s − 4·67-s + 14·73-s − 4·79-s + 12·83-s + 4·85-s + 6·89-s − 8·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.554·13-s − 0.485·17-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 1.35·35-s − 0.986·37-s − 0.312·41-s + 1.16·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s + 0.768·61-s + 0.496·65-s − 0.488·67-s + 1.63·73-s − 0.450·79-s + 1.31·83-s + 0.433·85-s + 0.635·89-s − 0.838·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.702728097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.702728097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.22177328277653, −13.63768051105924, −13.29785494262426, −12.41862318633261, −11.99218334705153, −11.76501969787956, −11.12838955408453, −10.74265927734064, −10.27193574437975, −9.544521747949240, −8.823411289482934, −8.497592345854381, −7.997131023836845, −7.542997754672149, −6.925800442028321, −6.568077150181307, −5.531754915911800, −5.077168766780158, −4.609264725612447, −4.164920214280098, −3.428230219726365, −2.651614261081221, −2.115681605588794, −1.182952898871035, −0.6160745001364024,
0.6160745001364024, 1.182952898871035, 2.115681605588794, 2.651614261081221, 3.428230219726365, 4.164920214280098, 4.609264725612447, 5.077168766780158, 5.531754915911800, 6.568077150181307, 6.925800442028321, 7.542997754672149, 7.997131023836845, 8.497592345854381, 8.823411289482934, 9.544521747949240, 10.27193574437975, 10.74265927734064, 11.12838955408453, 11.76501969787956, 11.99218334705153, 12.41862318633261, 13.29785494262426, 13.63768051105924, 14.22177328277653
