| L(s) = 1 | + 1.31i·5-s + 2.47·7-s − 35.9i·11-s + 65.5i·13-s + 87.1·17-s − 9.13i·19-s − 49.3·23-s + 123.·25-s − 122. i·29-s − 138.·31-s + 3.26i·35-s − 90.2i·37-s − 93.1·41-s + 191. i·43-s + 275.·47-s + ⋯ |
| L(s) = 1 | + 0.118i·5-s + 0.133·7-s − 0.986i·11-s + 1.39i·13-s + 1.24·17-s − 0.110i·19-s − 0.447·23-s + 0.986·25-s − 0.783i·29-s − 0.799·31-s + 0.0157i·35-s − 0.401i·37-s − 0.354·41-s + 0.679i·43-s + 0.856·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.208437194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208437194\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.31iT - 125T^{2} \) |
| 7 | \( 1 - 2.47T + 343T^{2} \) |
| 11 | \( 1 + 35.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 65.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 87.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 9.13iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 49.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 122. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 90.2iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 93.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 191. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 648. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 173. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 297. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 658. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 558.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 275.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 625.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 119. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 551.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 74.6T + 9.12e5T^{2} \) |
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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
−9.068133764926287248934755793594, −8.184056668443167457023583226912, −7.46713076487074849954985261407, −6.52317384320083207474670781817, −5.84939383550663040253921155778, −4.88906330854645700379617122971, −3.92236571216996782703470602387, −3.06747378681898511964038739112, −1.89422944193454267664512051983, −0.76782296741284559865438175220,
0.67265674838099762672079777976, 1.78069650660586663799308777770, 2.98087387497315378535050299655, 3.80946558963617037888079704823, 5.05618790012402005465352937937, 5.44236561236646034669756933704, 6.60481152160811779758684804768, 7.44509390666674134035382542996, 8.062826193985319157791938803005, 8.873186366063661628949249087430
