| L(s) = 1 | + 2.70·5-s − 7-s + 2.70·11-s + 0.0783·13-s − 4.34·17-s + 6.41·19-s − 2.34·23-s + 2.34·25-s − 1.78·29-s + 5.07·31-s − 2.70·35-s + 5.60·37-s + 12.6·41-s + 43-s + 10.3·47-s − 6·49-s − 4.68·53-s + 7.34·55-s + 5.65·59-s − 8.83·61-s + 0.212·65-s − 15.5·67-s + 12.6·71-s − 7.44·73-s − 2.70·77-s + 11.7·79-s + 2.21·83-s + ⋯ |
| L(s) = 1 | + 1.21·5-s − 0.377·7-s + 0.816·11-s + 0.0217·13-s − 1.05·17-s + 1.47·19-s − 0.487·23-s + 0.468·25-s − 0.331·29-s + 0.912·31-s − 0.457·35-s + 0.920·37-s + 1.98·41-s + 0.152·43-s + 1.50·47-s − 0.857·49-s − 0.642·53-s + 0.989·55-s + 0.736·59-s − 1.13·61-s + 0.0263·65-s − 1.89·67-s + 1.50·71-s − 0.871·73-s − 0.308·77-s + 1.32·79-s + 0.242·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.432561995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.432561995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 - 0.0783T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.68T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 7.44T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 2.21T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 9.34T + 97T^{2} \) |
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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
−9.057984622276509159862725505376, −7.894877667002562615875642780426, −7.13628085966715317110538251637, −6.14282833015698405458616622269, −5.99182744278261184587615894758, −4.86729505510619510572132712136, −4.02477646229827613295486120655, −2.92820680954515364799326968728, −2.07844023113000222295459912943, −0.989068742462070663431990036877,
0.989068742462070663431990036877, 2.07844023113000222295459912943, 2.92820680954515364799326968728, 4.02477646229827613295486120655, 4.86729505510619510572132712136, 5.99182744278261184587615894758, 6.14282833015698405458616622269, 7.13628085966715317110538251637, 7.894877667002562615875642780426, 9.057984622276509159862725505376
