| L(s) = 1 | − 2.38·2-s + 3.69·4-s + 3.31·7-s − 4.05·8-s + 4.36·11-s + 5.85·13-s − 7.91·14-s + 2.28·16-s − 0.407·17-s − 6.64·19-s − 10.4·22-s − 4.61·23-s − 13.9·26-s + 12.2·28-s − 3.30·29-s − 8.77·31-s + 2.66·32-s + 0.973·34-s + 3.09·37-s + 15.8·38-s − 2.89·41-s + 1.33·43-s + 16.1·44-s + 11.0·46-s + 11.0·47-s + 4.00·49-s + 21.6·52-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.84·4-s + 1.25·7-s − 1.43·8-s + 1.31·11-s + 1.62·13-s − 2.11·14-s + 0.570·16-s − 0.0989·17-s − 1.52·19-s − 2.22·22-s − 0.963·23-s − 2.74·26-s + 2.31·28-s − 0.613·29-s − 1.57·31-s + 0.470·32-s + 0.166·34-s + 0.509·37-s + 2.57·38-s − 0.451·41-s + 0.203·43-s + 2.43·44-s + 1.62·46-s + 1.61·47-s + 0.571·49-s + 3.00·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140035380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140035380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 + 0.407T + 17T^{2} \) |
| 19 | \( 1 + 6.64T + 19T^{2} \) |
| 23 | \( 1 + 4.61T + 23T^{2} \) |
| 29 | \( 1 + 3.30T + 29T^{2} \) |
| 31 | \( 1 + 8.77T + 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 - 5.80T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 0.0418T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 + 3.55T + 79T^{2} \) |
| 83 | \( 1 + 4.98T + 83T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 - 5.99T + 97T^{2} \) |
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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
−8.313910783899439557498521144028, −7.74398790635867650369067668085, −6.87001388117851618369966940251, −6.31304318064173506082046734013, −5.53906679254174943028904807831, −4.22373091832878594425787496270, −3.76388594574420382767800293977, −2.11219743741787562590723477683, −1.70332975067030845897347080759, −0.77758128342789692866125267086,
0.77758128342789692866125267086, 1.70332975067030845897347080759, 2.11219743741787562590723477683, 3.76388594574420382767800293977, 4.22373091832878594425787496270, 5.53906679254174943028904807831, 6.31304318064173506082046734013, 6.87001388117851618369966940251, 7.74398790635867650369067668085, 8.313910783899439557498521144028
