| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 2.61·11-s + 2.38·13-s + 2·14-s + 16-s − 0.381·17-s − 1.23·19-s + 2.61·22-s − 1.38·23-s + 2.38·26-s + 2·28-s − 2.09·29-s + 1.85·31-s + 32-s − 0.381·34-s + 11.0·37-s − 1.23·38-s + 8.94·41-s + 2.14·43-s + 2.61·44-s − 1.38·46-s − 8.09·47-s − 3·49-s + 2.38·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.755·7-s + 0.353·8-s + 0.789·11-s + 0.660·13-s + 0.534·14-s + 0.250·16-s − 0.0926·17-s − 0.283·19-s + 0.558·22-s − 0.288·23-s + 0.467·26-s + 0.377·28-s − 0.388·29-s + 0.333·31-s + 0.176·32-s − 0.0655·34-s + 1.82·37-s − 0.200·38-s + 1.39·41-s + 0.327·43-s + 0.394·44-s − 0.203·46-s − 1.18·47-s − 0.428·49-s + 0.330·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.368704817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.368704817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 + 0.381T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 - 2.14T + 43T^{2} \) |
| 47 | \( 1 + 8.09T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 + 6.18T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.033103953711349664997362676531, −8.085961097679328012973417493933, −7.54471715648565076245602203288, −6.38056841587281445965663345661, −6.03245771794791876663743147656, −4.87265009642107072646426849505, −4.26782664809766610198289168178, −3.39746396253606147142484798033, −2.22779649703611777853485168606, −1.19001480141886390193466526154,
1.19001480141886390193466526154, 2.22779649703611777853485168606, 3.39746396253606147142484798033, 4.26782664809766610198289168178, 4.87265009642107072646426849505, 6.03245771794791876663743147656, 6.38056841587281445965663345661, 7.54471715648565076245602203288, 8.085961097679328012973417493933, 9.033103953711349664997362676531
