| L(s) = 1 | − 0.794·3-s + 2.47·7-s − 2.36·9-s − 2.29·11-s − 3.84·13-s − 7.74·17-s − 2.29·19-s − 1.96·21-s − 23-s + 4.26·27-s + 5.28·29-s − 6.40·31-s + 1.82·33-s + 8.56·37-s + 3.05·39-s + 4.27·41-s − 1.88·43-s + 12.3·47-s − 0.889·49-s + 6.15·51-s − 7.57·53-s + 1.82·57-s + 6.07·59-s − 0.635·61-s − 5.85·63-s + 11.1·67-s + 0.794·69-s + ⋯ |
| L(s) = 1 | − 0.458·3-s + 0.934·7-s − 0.789·9-s − 0.693·11-s − 1.06·13-s − 1.87·17-s − 0.527·19-s − 0.428·21-s − 0.208·23-s + 0.821·27-s + 0.981·29-s − 1.14·31-s + 0.318·33-s + 1.40·37-s + 0.488·39-s + 0.667·41-s − 0.288·43-s + 1.80·47-s − 0.127·49-s + 0.862·51-s − 1.04·53-s + 0.242·57-s + 0.790·59-s − 0.0813·61-s − 0.737·63-s + 1.36·67-s + 0.0956·69-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
| \(L(1)\) |
\(\approx\) |
\(1.030344264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030344264\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.794T + 3T^{2} \) |
| 7 | \( 1 - 2.47T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 3.84T + 13T^{2} \) |
| 17 | \( 1 + 7.74T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 31 | \( 1 + 6.40T + 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 1.88T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 7.57T + 53T^{2} \) |
| 59 | \( 1 - 6.07T + 59T^{2} \) |
| 61 | \( 1 + 0.635T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 0.335T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 5.55T + 89T^{2} \) |
| 97 | \( 1 - 6.42T + 97T^{2} \) |
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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
−8.233304456568766993319744945058, −7.71816327968451243182689461686, −6.78832685433913313051253475037, −6.15620447667118786163399060537, −5.18115716494846645001023078658, −4.82745333768920034223328865853, −3.97460475904813273782382496982, −2.54364560648639716520477317487, −2.19036557682398692104585216422, −0.54656808089251075008767709253,
0.54656808089251075008767709253, 2.19036557682398692104585216422, 2.54364560648639716520477317487, 3.97460475904813273782382496982, 4.82745333768920034223328865853, 5.18115716494846645001023078658, 6.15620447667118786163399060537, 6.78832685433913313051253475037, 7.71816327968451243182689461686, 8.233304456568766993319744945058
