| L(s) = 1 | − 2.30i·3-s + 0.586i·5-s + (2.63 + 0.279i)7-s − 2.32·9-s + (0.922 + 3.18i)11-s − 13-s + 1.35·15-s − 3.52·17-s − 8.50·19-s + (0.645 − 6.07i)21-s − 4.27·23-s + 4.65·25-s − 1.55i·27-s + 7.21i·29-s + 8.28i·31-s + ⋯ |
| L(s) = 1 | − 1.33i·3-s + 0.262i·5-s + (0.994 + 0.105i)7-s − 0.775·9-s + (0.277 + 0.960i)11-s − 0.277·13-s + 0.349·15-s − 0.855·17-s − 1.95·19-s + (0.140 − 1.32i)21-s − 0.892·23-s + 0.931·25-s − 0.298i·27-s + 1.34i·29-s + 1.48i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5685226669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5685226669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.63 - 0.279i)T \) |
| 11 | \( 1 + (-0.922 - 3.18i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.30iT - 3T^{2} \) |
| 5 | \( 1 - 0.586iT - 5T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 + 8.50T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 - 7.21iT - 29T^{2} \) |
| 31 | \( 1 - 8.28iT - 31T^{2} \) |
| 37 | \( 1 - 4.64T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 8.25iT - 43T^{2} \) |
| 47 | \( 1 - 0.348iT - 47T^{2} \) |
| 53 | \( 1 + 8.42T + 53T^{2} \) |
| 59 | \( 1 - 10.8iT - 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 4.54T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−8.649260999444117603397734966604, −7.80403353189836840793150113088, −7.03752078131115948624451602057, −6.75946799063741367324984392928, −5.91804548471784819089998519664, −4.77493510759402031513109687069, −4.32872202993531818605992909065, −2.88599303669261807860452899394, −1.89816405419478862340246184972, −1.56703613008701100981932395215,
0.14766978231826844596366317202, 1.75539951696534321741282829242, 2.78483368281145642090495673254, 4.01432845661957203787746814435, 4.36269397192588243987674267376, 4.95985598805954387893939064192, 5.98792553251464323882615964375, 6.52910433917960008385798617109, 7.899247506796371994189726663021, 8.285049280282022075274689794104
