| L(s) = 1 | − 1.61·2-s + 0.618·4-s − 2·7-s + 2.23·8-s + 3·11-s − 13-s + 3.23·14-s − 4.85·16-s + 4.23·17-s − 6.70·19-s − 4.85·22-s − 5.38·23-s + 1.61·26-s − 1.23·28-s + 3.61·29-s + 8.70·31-s + 3.38·32-s − 6.85·34-s − 2·37-s + 10.8·38-s + 9.38·41-s − 7.38·43-s + 1.85·44-s + 8.70·46-s + 4.76·47-s − 3·49-s − 0.618·52-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s − 0.755·7-s + 0.790·8-s + 0.904·11-s − 0.277·13-s + 0.864·14-s − 1.21·16-s + 1.02·17-s − 1.53·19-s − 1.03·22-s − 1.12·23-s + 0.317·26-s − 0.233·28-s + 0.671·29-s + 1.56·31-s + 0.597·32-s − 1.17·34-s − 0.328·37-s + 1.76·38-s + 1.46·41-s − 1.12·43-s + 0.279·44-s + 1.28·46-s + 0.694·47-s − 0.428·49-s − 0.0857·52-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 9.38T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 9.14T + 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.058761387913769390759206754715, −7.18944872473669821507225983523, −6.45184662222692698774381120346, −5.97871628363432092026064896772, −4.68400757169133059369707768368, −4.14178004347461465395548155382, −3.14195708488296106043979789671, −2.08136400647833211258189485686, −1.09820080009033362205797219087, 0,
1.09820080009033362205797219087, 2.08136400647833211258189485686, 3.14195708488296106043979789671, 4.14178004347461465395548155382, 4.68400757169133059369707768368, 5.97871628363432092026064896772, 6.45184662222692698774381120346, 7.18944872473669821507225983523, 8.058761387913769390759206754715
