| L(s) = 1 | + 1.39·3-s − 5-s − 1.65·7-s − 1.05·9-s − 6.36·13-s − 1.39·15-s + 5.31·17-s + 4.36·19-s − 2.31·21-s − 5.57·23-s + 25-s − 5.65·27-s + 6.79·29-s + 0.259·31-s + 1.65·35-s − 0.791·37-s − 8.88·39-s + 2.74·41-s + 11.6·43-s + 1.05·45-s + 7.49·47-s − 4.25·49-s + 7.41·51-s − 1.84·53-s + 6.08·57-s − 7.15·59-s − 8.57·61-s + ⋯ |
| L(s) = 1 | + 0.805·3-s − 0.447·5-s − 0.625·7-s − 0.350·9-s − 1.76·13-s − 0.360·15-s + 1.28·17-s + 1.00·19-s − 0.504·21-s − 1.16·23-s + 0.200·25-s − 1.08·27-s + 1.26·29-s + 0.0466·31-s + 0.279·35-s − 0.130·37-s − 1.42·39-s + 0.427·41-s + 1.77·43-s + 0.156·45-s + 1.09·47-s − 0.608·49-s + 1.03·51-s − 0.253·53-s + 0.806·57-s − 0.931·59-s − 1.09·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.717287251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717287251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.39T + 3T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 13 | \( 1 + 6.36T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 23 | \( 1 + 5.57T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 - 0.259T + 31T^{2} \) |
| 37 | \( 1 + 0.791T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 7.49T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 1.05T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.27T + 79T^{2} \) |
| 83 | \( 1 - 7.15T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.971209564854642464024746028063, −7.80474762819578994419619564094, −7.06582462887833027311672706767, −6.07147118313015576278227455410, −5.34737207612494514156354633822, −4.47686917284949731896477669517, −3.52137131227300647344973421181, −2.92869138242575932648760063511, −2.21961216090076290228101373042, −0.66512922064585903616981670875,
0.66512922064585903616981670875, 2.21961216090076290228101373042, 2.92869138242575932648760063511, 3.52137131227300647344973421181, 4.47686917284949731896477669517, 5.34737207612494514156354633822, 6.07147118313015576278227455410, 7.06582462887833027311672706767, 7.80474762819578994419619564094, 7.971209564854642464024746028063
