| L(s) = 1 | − 3-s + 1.36·7-s + 9-s − 5.64·11-s + 13-s − 2.14·17-s + 2.41·19-s − 1.36·21-s + 2.72·23-s − 27-s − 5.28·29-s − 1.64·31-s + 5.64·33-s − 3.86·37-s − 39-s + 5.86·41-s + 9.00·43-s − 3.77·47-s − 5.14·49-s + 2.14·51-s − 12.0·53-s − 2.41·57-s + 6.19·59-s + 11.5·61-s + 1.36·63-s + 0.324·67-s − 2.72·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.515·7-s + 0.333·9-s − 1.70·11-s + 0.277·13-s − 0.519·17-s + 0.553·19-s − 0.297·21-s + 0.568·23-s − 0.192·27-s − 0.980·29-s − 0.295·31-s + 0.982·33-s − 0.635·37-s − 0.160·39-s + 0.916·41-s + 1.37·43-s − 0.551·47-s − 0.734·49-s + 0.299·51-s − 1.64·53-s − 0.319·57-s + 0.806·59-s + 1.48·61-s + 0.171·63-s + 0.0396·67-s − 0.328·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.244897699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244897699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 - 2.41T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 0.324T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 8.28T + 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.73392864804156021734863538023, −7.29944867645656172760258520513, −6.42371745310255394679206949906, −5.58505679949814194761449734491, −5.17324013302925341961322460105, −4.50360210309540243965134875292, −3.53551924642668006880083543848, −2.61678900211238879360640326458, −1.76774773119575064710442119044, −0.56446567111075060653312758195,
0.56446567111075060653312758195, 1.76774773119575064710442119044, 2.61678900211238879360640326458, 3.53551924642668006880083543848, 4.50360210309540243965134875292, 5.17324013302925341961322460105, 5.58505679949814194761449734491, 6.42371745310255394679206949906, 7.29944867645656172760258520513, 7.73392864804156021734863538023
