| L(s) = 1 | − 5.61·3-s + 5·5-s + 31.5·7-s + 4.47·9-s − 1.28·11-s + 22.9·13-s − 28.0·15-s + 72.5·17-s − 83.9·19-s − 176.·21-s − 23·23-s + 25·25-s + 126.·27-s − 276.·29-s + 226.·31-s + 7.19·33-s + 157.·35-s − 237.·37-s − 129.·39-s + 418.·41-s − 463.·43-s + 22.3·45-s + 427.·47-s + 652.·49-s − 407.·51-s + 93.0·53-s − 6.41·55-s + ⋯ |
| L(s) = 1 | − 1.07·3-s + 0.447·5-s + 1.70·7-s + 0.165·9-s − 0.0351·11-s + 0.490·13-s − 0.482·15-s + 1.03·17-s − 1.01·19-s − 1.83·21-s − 0.208·23-s + 0.200·25-s + 0.900·27-s − 1.76·29-s + 1.31·31-s + 0.0379·33-s + 0.761·35-s − 1.05·37-s − 0.529·39-s + 1.59·41-s − 1.64·43-s + 0.0741·45-s + 1.32·47-s + 1.90·49-s − 1.11·51-s + 0.241·53-s − 0.0157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.724225251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724225251\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 5.61T + 27T^{2} \) |
| 7 | \( 1 - 31.5T + 343T^{2} \) |
| 11 | \( 1 + 1.28T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 83.9T + 6.85e3T^{2} \) |
| 29 | \( 1 + 276.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 226.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 237.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 418.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 427.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 93.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 768.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 883.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 797.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 357.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 536.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−10.84805435030126915287753249771, −10.03625470502710903073513382015, −8.681204604981786401631558240878, −7.987629009212131687185431399212, −6.78926970694117940298996477308, −5.65579850102668702495493224426, −5.20666177529444469121715019019, −4.02629080439980003748035666079, −2.12375553848481785556077138695, −0.921058852467073143159510918342,
0.921058852467073143159510918342, 2.12375553848481785556077138695, 4.02629080439980003748035666079, 5.20666177529444469121715019019, 5.65579850102668702495493224426, 6.78926970694117940298996477308, 7.987629009212131687185431399212, 8.681204604981786401631558240878, 10.03625470502710903073513382015, 10.84805435030126915287753249771
