| L(s) = 1 | + 0.865i·5-s − i·7-s + 5.82·11-s + 0.727·13-s − 7.62i·17-s + 2.29i·19-s + 7.50·23-s + 4.25·25-s − 0.231i·29-s − 10.7i·31-s + 0.865·35-s + 1.64·37-s + 0.229i·41-s + 6.70i·43-s − 11.6·47-s + ⋯ |
| L(s) = 1 | + 0.386i·5-s − 0.377i·7-s + 1.75·11-s + 0.201·13-s − 1.84i·17-s + 0.527i·19-s + 1.56·23-s + 0.850·25-s − 0.0429i·29-s − 1.92i·31-s + 0.146·35-s + 0.270·37-s + 0.0358i·41-s + 1.02i·43-s − 1.70·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.328112662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328112662\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.865iT - 5T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 13 | \( 1 - 0.727T + 13T^{2} \) |
| 17 | \( 1 + 7.62iT - 17T^{2} \) |
| 19 | \( 1 - 2.29iT - 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 + 0.231iT - 29T^{2} \) |
| 31 | \( 1 + 10.7iT - 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 0.229iT - 41T^{2} \) |
| 43 | \( 1 - 6.70iT - 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2.80iT - 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 9.08T + 61T^{2} \) |
| 67 | \( 1 + 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 9.76T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 4.39iT - 79T^{2} \) |
| 83 | \( 1 + 2.47T + 83T^{2} \) |
| 89 | \( 1 - 3.15iT - 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−7.79740502115454008431676633740, −7.26489154769507433192067507699, −6.54329687229683115633841867553, −6.11496935687281450776600080520, −4.91355668258385105229143895466, −4.42976738901850107762128295387, −3.41745970786968255091356990791, −2.86944037053340666018344284402, −1.58598546962607878082735207403, −0.68684811294250356046854372767,
1.14188610530649318390707287256, 1.68505302160854665380923763120, 3.06323124541351269783937355723, 3.69556134602956781467998413233, 4.59157315156323187613638745865, 5.19959584435611673545448825707, 6.24736018428557353310670165614, 6.59545615168395068076227624822, 7.34599053799120460679712396893, 8.578965396663654528362227545980
