| L(s) = 1 | + 2·5-s − 3·9-s + 6·13-s − 25-s + 10·29-s + 2·37-s − 10·41-s − 6·45-s − 7·49-s + 14·53-s + 10·61-s + 12·65-s + 6·73-s + 9·81-s + 10·89-s − 18·97-s − 2·101-s − 6·109-s + 14·113-s − 18·117-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s + 1.66·13-s − 1/5·25-s + 1.85·29-s + 0.328·37-s − 1.56·41-s − 0.894·45-s − 49-s + 1.92·53-s + 1.28·61-s + 1.48·65-s + 0.702·73-s + 81-s + 1.05·89-s − 1.82·97-s − 0.199·101-s − 0.574·109-s + 1.31·113-s − 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.543769471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543769471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.937353973550093122096162853175, −6.66581854102807397515051773100, −6.44396028229397801847944387133, −5.63798887572403692287575825689, −5.20908074100741797334963082083, −4.15287533324461415156080717684, −3.36630586693417719338723595309, −2.62835145533620223272876983910, −1.73809998408379967764156075543, −0.78590544656973920591081333740,
0.78590544656973920591081333740, 1.73809998408379967764156075543, 2.62835145533620223272876983910, 3.36630586693417719338723595309, 4.15287533324461415156080717684, 5.20908074100741797334963082083, 5.63798887572403692287575825689, 6.44396028229397801847944387133, 6.66581854102807397515051773100, 7.937353973550093122096162853175
