| L(s) = 1 | + 2.18·3-s − 3.62·5-s + 4.31·7-s + 1.79·9-s − 11-s − 6.10·13-s − 7.94·15-s − 0.694·17-s − 19-s + 9.43·21-s + 7.79·23-s + 8.16·25-s − 2.64·27-s + 6.02·29-s + 6.26·31-s − 2.18·33-s − 15.6·35-s + 10.1·37-s − 13.3·39-s − 2.04·41-s + 2.16·43-s − 6.49·45-s + 10.2·47-s + 11.5·49-s − 1.52·51-s − 1.95·53-s + 3.62·55-s + ⋯ |
| L(s) = 1 | + 1.26·3-s − 1.62·5-s + 1.62·7-s + 0.596·9-s − 0.301·11-s − 1.69·13-s − 2.05·15-s − 0.168·17-s − 0.229·19-s + 2.05·21-s + 1.62·23-s + 1.63·25-s − 0.509·27-s + 1.11·29-s + 1.12·31-s − 0.380·33-s − 2.64·35-s + 1.66·37-s − 2.13·39-s − 0.319·41-s + 0.330·43-s − 0.968·45-s + 1.48·47-s + 1.65·49-s − 0.212·51-s − 0.268·53-s + 0.489·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.404084597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.404084597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 17 | \( 1 + 0.694T + 17T^{2} \) |
| 23 | \( 1 - 7.79T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 2.04T + 41T^{2} \) |
| 43 | \( 1 - 2.16T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 - 6.84T + 61T^{2} \) |
| 67 | \( 1 - 8.35T + 67T^{2} \) |
| 71 | \( 1 + 0.566T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 2.89T + 97T^{2} \) |
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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
−8.411658056582374857175593789491, −7.86802995798784309632482867945, −7.58961985314367556810403644122, −6.80690358868671074977552684005, −5.14787505303834535911330916691, −4.63793084648459265287550053949, −4.02345283148272429690748755803, −2.85927969076945514074356729167, −2.39184867897748383865875444586, −0.877402217181564622264380039451,
0.877402217181564622264380039451, 2.39184867897748383865875444586, 2.85927969076945514074356729167, 4.02345283148272429690748755803, 4.63793084648459265287550053949, 5.14787505303834535911330916691, 6.80690358868671074977552684005, 7.58961985314367556810403644122, 7.86802995798784309632482867945, 8.411658056582374857175593789491
