| L(s) = 1 | + 2.56·3-s − 5.12·7-s + 3.56·9-s + 11-s − 3.12·13-s − 2·17-s + 4·19-s − 13.1·21-s + 6.56·23-s + 1.43·27-s + 3.12·29-s + 1.43·31-s + 2.56·33-s + 3.43·37-s − 8·39-s + 7.12·41-s + 1.12·43-s + 8·47-s + 19.2·49-s − 5.12·51-s + 4.24·53-s + 10.2·57-s + 12.8·59-s − 7.12·61-s − 18.2·63-s + 5.43·67-s + 16.8·69-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 1.93·7-s + 1.18·9-s + 0.301·11-s − 0.866·13-s − 0.485·17-s + 0.917·19-s − 2.86·21-s + 1.36·23-s + 0.276·27-s + 0.579·29-s + 0.258·31-s + 0.445·33-s + 0.565·37-s − 1.28·39-s + 1.11·41-s + 0.171·43-s + 1.16·47-s + 2.74·49-s − 0.717·51-s + 0.583·53-s + 1.35·57-s + 1.66·59-s − 0.912·61-s − 2.29·63-s + 0.664·67-s + 2.02·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.578085685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.578085685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 7.12T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + 9.68T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.569294575681776858219842774550, −7.50716993787751304033219123967, −7.10375252011538161002163025072, −6.39089513160018131270292765758, −5.44481374385695821500459405959, −4.30205099804158779536748918872, −3.56694046126998976573294062261, −2.82462795893967783044342484210, −2.45923839405037145469780008008, −0.821154470808708518466482035081,
0.821154470808708518466482035081, 2.45923839405037145469780008008, 2.82462795893967783044342484210, 3.56694046126998976573294062261, 4.30205099804158779536748918872, 5.44481374385695821500459405959, 6.39089513160018131270292765758, 7.10375252011538161002163025072, 7.50716993787751304033219123967, 8.569294575681776858219842774550
