| L(s) = 1 | + 3-s − 7-s + 9-s − 5.62·11-s − 2.57·13-s + 6.20·17-s − 19-s − 21-s + 7.83·23-s − 5·25-s + 27-s − 3.42·29-s + 5.04·31-s − 5.62·33-s − 7.04·37-s − 2.57·39-s − 9.25·41-s + 7.25·43-s + 3.62·47-s + 49-s + 6.20·51-s − 8.57·53-s − 57-s − 8.41·59-s − 14.4·61-s − 63-s − 12.8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 0.333·9-s − 1.69·11-s − 0.715·13-s + 1.50·17-s − 0.229·19-s − 0.218·21-s + 1.63·23-s − 25-s + 0.192·27-s − 0.635·29-s + 0.906·31-s − 0.979·33-s − 1.15·37-s − 0.412·39-s − 1.44·41-s + 1.10·43-s + 0.529·47-s + 0.142·49-s + 0.868·51-s − 1.17·53-s − 0.132·57-s − 1.09·59-s − 1.84·61-s − 0.125·63-s − 1.57·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 + 2.57T + 13T^{2} \) |
| 17 | \( 1 - 6.20T + 17T^{2} \) |
| 23 | \( 1 - 7.83T + 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 31 | \( 1 - 5.04T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 + 9.25T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 - 3.62T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 + 8.41T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.25T + 73T^{2} \) |
| 79 | \( 1 + 6.67T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 7.15T + 89T^{2} \) |
| 97 | \( 1 - 4.78T + 97T^{2} \) |
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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
−8.164359453464684338015892229815, −7.59861137563680160646433092771, −7.08675817711983935797107690471, −5.88781429319536879421774114884, −5.24247050852249481085549242203, −4.44726954790598706654071024306, −3.14028059375825214603687456599, −2.87256943672115723737366308386, −1.60213031466138144721619213968, 0,
1.60213031466138144721619213968, 2.87256943672115723737366308386, 3.14028059375825214603687456599, 4.44726954790598706654071024306, 5.24247050852249481085549242203, 5.88781429319536879421774114884, 7.08675817711983935797107690471, 7.59861137563680160646433092771, 8.164359453464684338015892229815
