| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4.89·11-s − 0.449·13-s − 14-s + 16-s − 2·17-s + 6.44·19-s − 4.89·22-s + 6.89·23-s − 0.449·26-s − 28-s + 2.89·29-s − 0.898·31-s + 32-s − 2·34-s + 2·37-s + 6.44·38-s + 10.8·41-s + 8.89·43-s − 4.89·44-s + 6.89·46-s + 0.898·47-s + 49-s − 0.449·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.377·7-s + 0.353·8-s − 1.47·11-s − 0.124·13-s − 0.267·14-s + 0.250·16-s − 0.485·17-s + 1.47·19-s − 1.04·22-s + 1.43·23-s − 0.0881·26-s − 0.188·28-s + 0.538·29-s − 0.161·31-s + 0.176·32-s − 0.342·34-s + 0.328·37-s + 1.04·38-s + 1.70·41-s + 1.35·43-s − 0.738·44-s + 1.01·46-s + 0.131·47-s + 0.142·49-s − 0.0623·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.663835451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.663835451\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 + 0.898T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 - 0.898T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 - 8.44T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 + 2.89T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 3.79T + 97T^{2} \) |
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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
−8.671501638574013276719841265501, −7.60042568897851394711821255594, −7.30888466539120662197892038024, −6.30278614976401395409516296359, −5.45890174961157953891347456045, −4.97905503640449219249066760321, −4.00625611932096696610024124243, −2.94152371679574146245178227398, −2.49277899131395804471994601550, −0.886049184299352588669362709967,
0.886049184299352588669362709967, 2.49277899131395804471994601550, 2.94152371679574146245178227398, 4.00625611932096696610024124243, 4.97905503640449219249066760321, 5.45890174961157953891347456045, 6.30278614976401395409516296359, 7.30888466539120662197892038024, 7.60042568897851394711821255594, 8.671501638574013276719841265501
