| L(s) = 1 | − 3.04i·5-s − 0.418·7-s + 3·9-s − 5.27i·11-s + 2.62i·13-s + 3.27·17-s − i·19-s − 3.46·23-s − 4.27·25-s + 2.62i·29-s − 6.09·31-s + 1.27i·35-s − 9.55i·37-s + 4.54·41-s − 2.72i·43-s + ⋯ |
| L(s) = 1 | − 1.36i·5-s − 0.158·7-s + 9-s − 1.59i·11-s + 0.728i·13-s + 0.794·17-s − 0.229i·19-s − 0.722·23-s − 0.854·25-s + 0.487i·29-s − 1.09·31-s + 0.215i·35-s − 1.57i·37-s + 0.710·41-s − 0.415i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.550361323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550361323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 3.04iT - 5T^{2} \) |
| 7 | \( 1 + 0.418T + 7T^{2} \) |
| 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 2.62iT - 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 + 9.55iT - 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 + 2.72iT - 43T^{2} \) |
| 47 | \( 1 - 0.418T + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 6.54iT - 59T^{2} \) |
| 61 | \( 1 + 3.04iT - 61T^{2} \) |
| 67 | \( 1 - 6.54iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 17.0iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−9.300117005932263488742384645401, −8.807849970460476445880146326983, −7.981639441508249745553963450590, −7.09177050017374049593318109586, −5.96514610925549358865688515524, −5.28208152859081322757927669281, −4.26495244049795692351410865526, −3.48783627673262881550463045916, −1.80386150065715661652496502166, −0.68827073265550031454047029351,
1.67657201673385603884662542277, 2.80040268024149812142606011721, 3.79522109804007823105823398979, 4.75296332021822054274695488414, 5.94565955816806508714225882312, 6.81404556620794821987041429226, 7.42605528874647181653680316391, 8.017362304981690183102662729671, 9.611321305326071882233976309459, 9.942334024131057043875128936318
