| L(s) = 1 | + 2.45·5-s − 5.04·7-s − 3·9-s − 2.96·11-s − 6.51·13-s − 6.41·17-s + 19-s + 7.57·23-s + 1.00·25-s + 9.36·29-s − 6.09·31-s − 12.3·35-s − 0.659·37-s + 4.13·41-s − 1.33·43-s − 7.35·45-s − 11.5·47-s + 18.4·49-s − 1.34·53-s − 7.27·55-s + 5.20·59-s + 8.44·61-s + 15.1·63-s − 15.9·65-s − 4.86·67-s + 2.09·71-s + 5.55·73-s + ⋯ |
| L(s) = 1 | + 1.09·5-s − 1.90·7-s − 9-s − 0.895·11-s − 1.80·13-s − 1.55·17-s + 0.229·19-s + 1.58·23-s + 0.201·25-s + 1.73·29-s − 1.09·31-s − 2.08·35-s − 0.108·37-s + 0.645·41-s − 0.203·43-s − 1.09·45-s − 1.67·47-s + 2.63·49-s − 0.184·53-s − 0.981·55-s + 0.678·59-s + 1.08·61-s + 1.90·63-s − 1.98·65-s − 0.594·67-s + 0.248·71-s + 0.649·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7223522926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7223522926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 + 6.51T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 - 9.36T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 + 0.659T + 37T^{2} \) |
| 41 | \( 1 - 4.13T + 41T^{2} \) |
| 43 | \( 1 + 1.33T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 - 5.20T + 59T^{2} \) |
| 61 | \( 1 - 8.44T + 61T^{2} \) |
| 67 | \( 1 + 4.86T + 67T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 83 | \( 1 + 1.15T + 83T^{2} \) |
| 89 | \( 1 - 7.00T + 89T^{2} \) |
| 97 | \( 1 + 8.42T + 97T^{2} \) |
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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
−8.167136186978467331820739534297, −6.94954312289246028149038333910, −6.82288892709415646066982542321, −5.94935483529131448402272422238, −5.29626959349422490394714362522, −4.67812131794134052987014824986, −3.29275946251221900391537076247, −2.67392106407307981181971101220, −2.28029592204728095805483094219, −0.40470571370775683434275271680,
0.40470571370775683434275271680, 2.28029592204728095805483094219, 2.67392106407307981181971101220, 3.29275946251221900391537076247, 4.67812131794134052987014824986, 5.29626959349422490394714362522, 5.94935483529131448402272422238, 6.82288892709415646066982542321, 6.94954312289246028149038333910, 8.167136186978467331820739534297
