| L(s) = 1 | + 3-s − 0.419·5-s + 4.23·7-s + 9-s + 2.07·11-s − 5.80·13-s − 0.419·15-s − 0.0751·17-s + 3.35·19-s + 4.23·21-s − 4.32·23-s − 4.82·25-s + 27-s + 8.99·29-s − 6.91·31-s + 2.07·33-s − 1.77·35-s + 6.31·37-s − 5.80·39-s + 4.28·41-s + 1.06·43-s − 0.419·45-s + 9.61·47-s + 10.9·49-s − 0.0751·51-s − 14.0·53-s − 0.868·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.187·5-s + 1.59·7-s + 0.333·9-s + 0.624·11-s − 1.61·13-s − 0.108·15-s − 0.0182·17-s + 0.769·19-s + 0.923·21-s − 0.901·23-s − 0.964·25-s + 0.192·27-s + 1.66·29-s − 1.24·31-s + 0.360·33-s − 0.299·35-s + 1.03·37-s − 0.930·39-s + 0.669·41-s + 0.162·43-s − 0.0624·45-s + 1.40·47-s + 1.55·49-s − 0.0105·51-s − 1.93·53-s − 0.117·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.074158676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.074158676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.419T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 + 5.80T + 13T^{2} \) |
| 17 | \( 1 + 0.0751T + 17T^{2} \) |
| 19 | \( 1 - 3.35T + 19T^{2} \) |
| 23 | \( 1 + 4.32T + 23T^{2} \) |
| 29 | \( 1 - 8.99T + 29T^{2} \) |
| 31 | \( 1 + 6.91T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 - 1.06T + 43T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 3.20T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 5.13T + 67T^{2} \) |
| 71 | \( 1 - 0.458T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 4.93T + 79T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.79735738321118416672067821226, −7.47959345308838123834540307635, −6.57951865517465475371561474418, −5.61280609723810248122950553840, −4.87087234241025747650018529991, −4.37401725019044068814799874750, −3.59055507338738548288004755710, −2.45273146325775411071043707195, −1.94563882939919367697760155809, −0.864293834673023153880809109299,
0.864293834673023153880809109299, 1.94563882939919367697760155809, 2.45273146325775411071043707195, 3.59055507338738548288004755710, 4.37401725019044068814799874750, 4.87087234241025747650018529991, 5.61280609723810248122950553840, 6.57951865517465475371561474418, 7.47959345308838123834540307635, 7.79735738321118416672067821226
