| L(s) = 1 | − 2.61·2-s + 4.81·4-s − 1.21·7-s − 7.35·8-s + 5.39·11-s + 4.87·13-s + 3.15·14-s + 9.57·16-s − 0.980·17-s + 3.51·19-s − 14.0·22-s − 2.29·23-s − 12.7·26-s − 5.82·28-s − 5.70·29-s + 8.17·31-s − 10.2·32-s + 2.56·34-s + 5.75·37-s − 9.17·38-s + 9.10·41-s + 5.36·43-s + 26.0·44-s + 6.00·46-s + 1.03·47-s − 5.53·49-s + 23.4·52-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 2.40·4-s − 0.457·7-s − 2.60·8-s + 1.62·11-s + 1.35·13-s + 0.844·14-s + 2.39·16-s − 0.237·17-s + 0.806·19-s − 3.00·22-s − 0.479·23-s − 2.49·26-s − 1.10·28-s − 1.05·29-s + 1.46·31-s − 1.81·32-s + 0.439·34-s + 0.945·37-s − 1.48·38-s + 1.42·41-s + 0.818·43-s + 3.92·44-s + 0.885·46-s + 0.150·47-s − 0.790·49-s + 3.25·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012814542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012814542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 - 5.39T + 11T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 + 0.980T + 17T^{2} \) |
| 19 | \( 1 - 3.51T + 19T^{2} \) |
| 23 | \( 1 + 2.29T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 - 8.17T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 + 2.12T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 - 9.98T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.177T + 71T^{2} \) |
| 73 | \( 1 + 5.33T + 73T^{2} \) |
| 79 | \( 1 + 4.81T + 79T^{2} \) |
| 83 | \( 1 - 4.01T + 83T^{2} \) |
| 89 | \( 1 - 2.08T + 89T^{2} \) |
| 97 | \( 1 + 6.78T + 97T^{2} \) |
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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
−8.272083263410287583697165353324, −7.63843144879584099822471442926, −6.81469337716660686610801765939, −6.31109670039716804319845760680, −5.79577589583785826640699001019, −4.21541325264072871043910321453, −3.47461727280684629358214628321, −2.47172824016447741525780671572, −1.40351727265331554295803377899, −0.800105883061123991043509187484,
0.800105883061123991043509187484, 1.40351727265331554295803377899, 2.47172824016447741525780671572, 3.47461727280684629358214628321, 4.21541325264072871043910321453, 5.79577589583785826640699001019, 6.31109670039716804319845760680, 6.81469337716660686610801765939, 7.63843144879584099822471442926, 8.272083263410287583697165353324
