L(s) = 1 | + 2.14·3-s + 1.14·7-s + 1.60·9-s + 5.89·11-s + 4.89·13-s + 5.89·17-s − 2.34·19-s + 2.45·21-s − 23-s − 3.00·27-s + 3.74·29-s + 5.68·31-s + 12.6·33-s − 4·37-s + 10.4·39-s − 1.05·41-s − 11.4·43-s − 7.74·47-s − 5.68·49-s + 12.6·51-s − 12.9·53-s − 5.03·57-s + 0.797·59-s + 13.8·61-s + 1.83·63-s + 15.5·67-s − 2.14·69-s + ⋯ |
L(s) = 1 | + 1.23·3-s + 0.432·7-s + 0.533·9-s + 1.77·11-s + 1.35·13-s + 1.42·17-s − 0.538·19-s + 0.536·21-s − 0.208·23-s − 0.577·27-s + 0.695·29-s + 1.02·31-s + 2.20·33-s − 0.657·37-s + 1.68·39-s − 0.165·41-s − 1.75·43-s − 1.12·47-s − 0.812·49-s + 1.76·51-s − 1.78·53-s − 0.667·57-s + 0.103·59-s + 1.77·61-s + 0.231·63-s + 1.90·67-s − 0.258·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.016589947\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.016589947\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2.14T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 5.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 5.89T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 0.797T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 2.94T + 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 0.912T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.312517420722612765444747844918, −8.014122844057299954646566086209, −6.75554615938553402989547835012, −6.39973442969171879463051454901, −5.35846383151651284548569911974, −4.31636282896027431775560284745, −3.58011863683475866890809637968, −3.14721774688353596138073608584, −1.81052369145788542817098728257, −1.21450282551685192395197972281,
1.21450282551685192395197972281, 1.81052369145788542817098728257, 3.14721774688353596138073608584, 3.58011863683475866890809637968, 4.31636282896027431775560284745, 5.35846383151651284548569911974, 6.39973442969171879463051454901, 6.75554615938553402989547835012, 8.014122844057299954646566086209, 8.312517420722612765444747844918