| L(s) = 1 | − 3-s + 1.41·5-s + 4.24·7-s + 9-s + 6·11-s + 5.65·13-s − 1.41·15-s − 6·17-s + 4·19-s − 4.24·21-s − 2.82·23-s − 2.99·25-s − 27-s + 1.41·29-s + 1.41·31-s − 6·33-s + 6·35-s − 8.48·37-s − 5.65·39-s − 2·41-s + 1.41·45-s − 2.82·47-s + 10.9·49-s + 6·51-s − 9.89·53-s + 8.48·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.632·5-s + 1.60·7-s + 0.333·9-s + 1.80·11-s + 1.56·13-s − 0.365·15-s − 1.45·17-s + 0.917·19-s − 0.925·21-s − 0.589·23-s − 0.599·25-s − 0.192·27-s + 0.262·29-s + 0.254·31-s − 1.04·33-s + 1.01·35-s − 1.39·37-s − 0.905·39-s − 0.312·41-s + 0.210·45-s − 0.412·47-s + 1.57·49-s + 0.840·51-s − 1.35·53-s + 1.14·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.218404552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.218404552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.329028849130003149968887434696, −8.729806734291771044247742700167, −7.969595087606173772397230191303, −6.76922470237093041071592885437, −6.25582842784896949155701324695, −5.36322584255679373857777577392, −4.45814122226574365449070356385, −3.70383039227769137582396865576, −1.84344958307031385167369536521, −1.30590714772851114620463181039,
1.30590714772851114620463181039, 1.84344958307031385167369536521, 3.70383039227769137582396865576, 4.45814122226574365449070356385, 5.36322584255679373857777577392, 6.25582842784896949155701324695, 6.76922470237093041071592885437, 7.969595087606173772397230191303, 8.729806734291771044247742700167, 9.329028849130003149968887434696
