| L(s) = 1 | + 1.32·2-s − 3-s − 0.241·4-s + 0.903·5-s − 1.32·6-s − 2.97·8-s + 9-s + 1.19·10-s − 0.215·11-s + 0.241·12-s − 4.39·13-s − 0.903·15-s − 3.45·16-s − 7.88·17-s + 1.32·18-s + 6.94·19-s − 0.217·20-s − 0.285·22-s + 2.86·23-s + 2.97·24-s − 4.18·25-s − 5.82·26-s − 27-s − 4.84·29-s − 1.19·30-s + 4.36·31-s + 1.35·32-s + ⋯ |
| L(s) = 1 | + 0.937·2-s − 0.577·3-s − 0.120·4-s + 0.404·5-s − 0.541·6-s − 1.05·8-s + 0.333·9-s + 0.378·10-s − 0.0649·11-s + 0.0695·12-s − 1.21·13-s − 0.233·15-s − 0.864·16-s − 1.91·17-s + 0.312·18-s + 1.59·19-s − 0.0487·20-s − 0.0609·22-s + 0.598·23-s + 0.606·24-s − 0.836·25-s − 1.14·26-s − 0.192·27-s − 0.899·29-s − 0.218·30-s + 0.783·31-s + 0.239·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693571852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693571852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 5 | \( 1 - 0.903T + 5T^{2} \) |
| 11 | \( 1 + 0.215T + 11T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 17 | \( 1 + 7.88T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 - 4.36T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 0.643T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 0.978T + 71T^{2} \) |
| 73 | \( 1 + 9.48T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 9.10T + 83T^{2} \) |
| 89 | \( 1 - 6.38T + 89T^{2} \) |
| 97 | \( 1 - 9.92T + 97T^{2} \) |
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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
−7.84439452199581605740728558025, −7.22668740694177095181079419015, −6.38179677380893030444107099161, −5.84504831160075001007785007220, −5.04331189428368975979090134174, −4.67338265623784797252590651931, −3.86857069215370639046515940143, −2.84806103441397952326880847316, −2.10863459092279479335136020782, −0.58844262951274650979429816874,
0.58844262951274650979429816874, 2.10863459092279479335136020782, 2.84806103441397952326880847316, 3.86857069215370639046515940143, 4.67338265623784797252590651931, 5.04331189428368975979090134174, 5.84504831160075001007785007220, 6.38179677380893030444107099161, 7.22668740694177095181079419015, 7.84439452199581605740728558025
