| L(s) = 1 | − 0.848·3-s − 1.74·7-s − 2.28·9-s + 5.92·11-s − 6.78·13-s − 1.86·17-s − 19-s + 1.48·21-s − 5.94·23-s + 4.47·27-s + 3.29·29-s − 5.75·31-s − 5.02·33-s + 4.36·37-s + 5.75·39-s + 7.12·41-s + 6.98·43-s + 4.02·47-s − 3.95·49-s + 1.57·51-s + 9.19·53-s + 0.848·57-s + 2.51·59-s − 2.49·61-s + 3.97·63-s − 6.90·67-s + 5.04·69-s + ⋯ |
| L(s) = 1 | − 0.489·3-s − 0.659·7-s − 0.760·9-s + 1.78·11-s − 1.88·13-s − 0.451·17-s − 0.229·19-s + 0.322·21-s − 1.23·23-s + 0.862·27-s + 0.612·29-s − 1.03·31-s − 0.874·33-s + 0.717·37-s + 0.921·39-s + 1.11·41-s + 1.06·43-s + 0.587·47-s − 0.565·49-s + 0.220·51-s + 1.26·53-s + 0.112·57-s + 0.327·59-s − 0.319·61-s + 0.501·63-s − 0.843·67-s + 0.606·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9588920982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9588920982\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.848T + 3T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 6.78T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 + 5.75T + 31T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 - 4.02T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 - 2.51T + 59T^{2} \) |
| 61 | \( 1 + 2.49T + 61T^{2} \) |
| 67 | \( 1 + 6.90T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.684450401406013114131499073894, −7.58216666096238724875058967415, −6.97388978481289279810179985652, −6.17974203231934965052202009037, −5.74460954785962747724452208281, −4.61047569535495913284902192739, −4.02797270679690645626290552090, −2.92181657668714199430125623287, −2.05859302237289275087996841947, −0.55777705409897091411123674292,
0.55777705409897091411123674292, 2.05859302237289275087996841947, 2.92181657668714199430125623287, 4.02797270679690645626290552090, 4.61047569535495913284902192739, 5.74460954785962747724452208281, 6.17974203231934965052202009037, 6.97388978481289279810179985652, 7.58216666096238724875058967415, 8.684450401406013114131499073894
