| L(s) = 1 | + 2.67·2-s − 0.844·4-s + 15.4·7-s − 23.6·8-s − 44.5·11-s + 28.0·13-s + 41.2·14-s − 56.5·16-s + 92.6·17-s + 49.5·19-s − 119.·22-s − 0.922·23-s + 74.9·26-s − 13.0·28-s − 189.·29-s − 299.·31-s + 38.0·32-s + 247.·34-s + 57.8·37-s + 132.·38-s + 287.·41-s − 0.591·43-s + 37.6·44-s − 2.46·46-s + 598.·47-s − 105.·49-s − 23.6·52-s + ⋯ |
| L(s) = 1 | + 0.945·2-s − 0.105·4-s + 0.832·7-s − 1.04·8-s − 1.22·11-s + 0.597·13-s + 0.787·14-s − 0.883·16-s + 1.32·17-s + 0.598·19-s − 1.15·22-s − 0.00836·23-s + 0.565·26-s − 0.0878·28-s − 1.21·29-s − 1.73·31-s + 0.210·32-s + 1.25·34-s + 0.256·37-s + 0.566·38-s + 1.09·41-s − 0.00209·43-s + 0.128·44-s − 0.00791·46-s + 1.85·47-s − 0.307·49-s − 0.0630·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.076431389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.076431389\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.67T + 8T^{2} \) |
| 7 | \( 1 - 15.4T + 343T^{2} \) |
| 11 | \( 1 + 44.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.922T + 1.21e4T^{2} \) |
| 29 | \( 1 + 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 57.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 0.591T + 7.95e4T^{2} \) |
| 47 | \( 1 - 598.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 146.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 193.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 566.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 355.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 320.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 636.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 287.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 285.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 331.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−8.810662252316245513601367265930, −7.78928091872179907704921958790, −7.43363225733884480244331694943, −5.91826848254757701045739536472, −5.53176702975605724922881987385, −4.85920889056627064541851129878, −3.86766057849240537252707319569, −3.16463067656182180738446699691, −2.03573903856288053934517676877, −0.69687126605436575228234959587,
0.69687126605436575228234959587, 2.03573903856288053934517676877, 3.16463067656182180738446699691, 3.86766057849240537252707319569, 4.85920889056627064541851129878, 5.53176702975605724922881987385, 5.91826848254757701045739536472, 7.43363225733884480244331694943, 7.78928091872179907704921958790, 8.810662252316245513601367265930
