| L(s) = 1 | − 0.565·2-s − 3.09·3-s − 1.68·4-s + 1.80·5-s + 1.74·6-s + 2.08·8-s + 6.55·9-s − 1.02·10-s + 3.58·11-s + 5.19·12-s − 5.57·15-s + 2.18·16-s − 6.14·17-s − 3.70·18-s − 2.51·19-s − 3.03·20-s − 2.02·22-s + 4.13·23-s − 6.43·24-s − 1.74·25-s − 10.9·27-s + 5.11·29-s + 3.15·30-s + 4.55·31-s − 5.39·32-s − 11.0·33-s + 3.47·34-s + ⋯ |
| L(s) = 1 | − 0.399·2-s − 1.78·3-s − 0.840·4-s + 0.807·5-s + 0.713·6-s + 0.735·8-s + 2.18·9-s − 0.322·10-s + 1.08·11-s + 1.49·12-s − 1.44·15-s + 0.545·16-s − 1.49·17-s − 0.873·18-s − 0.576·19-s − 0.678·20-s − 0.432·22-s + 0.861·23-s − 1.31·24-s − 0.348·25-s − 2.11·27-s + 0.949·29-s + 0.576·30-s + 0.817·31-s − 0.954·32-s − 1.92·33-s + 0.596·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8280447633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8280447633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.565T + 2T^{2} \) |
| 3 | \( 1 + 3.09T + 3T^{2} \) |
| 5 | \( 1 - 1.80T + 5T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 1.86T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 - 2.02T + 53T^{2} \) |
| 59 | \( 1 - 9.10T + 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 - 5.20T + 79T^{2} \) |
| 83 | \( 1 - 9.42T + 83T^{2} \) |
| 89 | \( 1 - 0.355T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.73943501067920466776085039816, −6.73626121128354515862613690492, −6.45395550518911582227034917962, −5.80890315501814394888944758762, −5.01491614349664081152812963958, −4.47306981329543928444050879992, −3.95692315763771588756372646238, −2.34652631797941835251003261040, −1.28263997576113568731368638114, −0.61328165384748465096662051556,
0.61328165384748465096662051556, 1.28263997576113568731368638114, 2.34652631797941835251003261040, 3.95692315763771588756372646238, 4.47306981329543928444050879992, 5.01491614349664081152812963958, 5.80890315501814394888944758762, 6.45395550518911582227034917962, 6.73626121128354515862613690492, 7.73943501067920466776085039816
