| L(s) = 1 | + 2.64·2-s + 5.00·4-s − 3·7-s + 7.93·8-s + 5.29·11-s + 2·13-s − 7.93·14-s + 11.0·16-s − 5.29·17-s + 19-s + 14.0·22-s + 5.29·26-s − 15.0·28-s − 5.29·29-s − 3·31-s + 13.2·32-s − 14.0·34-s + 37-s + 2.64·38-s − 5.29·41-s − 43-s + 26.4·44-s − 5.29·47-s + 2·49-s + 10.0·52-s − 5.29·53-s − 23.8·56-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.50·4-s − 1.13·7-s + 2.80·8-s + 1.59·11-s + 0.554·13-s − 2.12·14-s + 2.75·16-s − 1.28·17-s + 0.229·19-s + 2.98·22-s + 1.03·26-s − 2.83·28-s − 0.982·29-s − 0.538·31-s + 2.33·32-s − 2.40·34-s + 0.164·37-s + 0.429·38-s − 0.826·41-s − 0.152·43-s + 3.98·44-s − 0.771·47-s + 0.285·49-s + 1.38·52-s − 0.726·53-s − 3.18·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.347878439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.347878439\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.64T + 2T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−11.01949562177065143244385887088, −9.753131323959798992367574928052, −8.854970762158158320656806156993, −7.30537812173058972741068848350, −6.45025569392483847593973729007, −6.16482571886967247338423844759, −4.86773569835033415573584131310, −3.85601571529966850519457451903, −3.30375076762567740142856987476, −1.87220090675730271137255508551,
1.87220090675730271137255508551, 3.30375076762567740142856987476, 3.85601571529966850519457451903, 4.86773569835033415573584131310, 6.16482571886967247338423844759, 6.45025569392483847593973729007, 7.30537812173058972741068848350, 8.854970762158158320656806156993, 9.753131323959798992367574928052, 11.01949562177065143244385887088
