| L(s) = 1 | + 15.8·5-s + 17.8·7-s − 52.9·11-s − 8.43·13-s − 129.·17-s + 50.4·19-s + 128.·23-s + 126.·25-s + 111.·29-s + 302.·31-s + 283.·35-s + 182.·37-s + 94.5·41-s + 184.·43-s + 296.·47-s − 24.1·49-s − 102.·53-s − 839.·55-s + 93.3·59-s + 338.·61-s − 133.·65-s + 489.·67-s − 86.9·71-s − 154.·73-s − 945.·77-s + 449.·79-s − 383.·83-s + ⋯ |
| L(s) = 1 | + 1.41·5-s + 0.964·7-s − 1.45·11-s − 0.179·13-s − 1.84·17-s + 0.609·19-s + 1.16·23-s + 1.01·25-s + 0.712·29-s + 1.75·31-s + 1.36·35-s + 0.813·37-s + 0.360·41-s + 0.654·43-s + 0.921·47-s − 0.0704·49-s − 0.266·53-s − 2.05·55-s + 0.206·59-s + 0.711·61-s − 0.255·65-s + 0.891·67-s − 0.145·71-s − 0.247·73-s − 1.39·77-s + 0.640·79-s − 0.507·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.985341033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.985341033\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 15.8T + 125T^{2} \) |
| 7 | \( 1 - 17.8T + 343T^{2} \) |
| 11 | \( 1 + 52.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.43T + 2.19e3T^{2} \) |
| 17 | \( 1 + 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 111.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 302.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 94.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 296.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 102.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 93.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 338.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 489.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 86.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 449.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 383.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 517.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.73e3T + 9.12e5T^{2} \) |
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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
−9.441459499161729544598997185917, −8.651910157887449987180556197540, −7.85479912013808595673996317229, −6.85663982471742886323434682254, −5.98136192580979019012121248397, −5.06944870754815578145752451427, −4.58891629868991337732050299903, −2.70864990701583260923350479167, −2.21298656889584361449449615550, −0.910587434348081480585361841018,
0.910587434348081480585361841018, 2.21298656889584361449449615550, 2.70864990701583260923350479167, 4.58891629868991337732050299903, 5.06944870754815578145752451427, 5.98136192580979019012121248397, 6.85663982471742886323434682254, 7.85479912013808595673996317229, 8.651910157887449987180556197540, 9.441459499161729544598997185917
