| L(s) = 1 | − 1.21i·2-s − 0.688i·3-s + 0.525·4-s − 0.836·6-s + i·7-s − 3.06i·8-s + 2.52·9-s − 11-s − 0.361i·12-s − 3.73i·13-s + 1.21·14-s − 2.67·16-s − 0.0666i·17-s − 3.06i·18-s − 6.42·19-s + ⋯ |
| L(s) = 1 | − 0.858i·2-s − 0.397i·3-s + 0.262·4-s − 0.341·6-s + 0.377i·7-s − 1.08i·8-s + 0.841·9-s − 0.301·11-s − 0.104i·12-s − 1.03i·13-s + 0.324·14-s − 0.668·16-s − 0.0161i·17-s − 0.722i·18-s − 1.47·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.860113133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860113133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.21iT - 2T^{2} \) |
| 3 | \( 1 + 0.688iT - 3T^{2} \) |
| 13 | \( 1 + 3.73iT - 13T^{2} \) |
| 17 | \( 1 + 0.0666iT - 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 23 | \( 1 + 1.09iT - 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 + 1.33iT - 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 + 11.7iT - 43T^{2} \) |
| 47 | \( 1 - 2.26iT - 47T^{2} \) |
| 53 | \( 1 + 1.71iT - 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 1.17iT - 73T^{2} \) |
| 79 | \( 1 - 8.51T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 13.4iT - 97T^{2} \) |
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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
−8.930379497331516487247665895244, −8.095061181370179363283226749698, −7.26265303384219352566633621923, −6.56393075307947034155698876738, −5.74082821238496693507751160272, −4.57861386687208136392413157453, −3.66960262361683473590021063174, −2.61015906477758604299990354707, −1.91343608918237307081486210320, −0.65243444739390920322467575428,
1.58130009588248500882066060616, 2.67693356506046579204979815500, 4.10065722091876943732271427225, 4.61591867009673525662927064466, 5.65981054917977604702061472907, 6.61546470258124734656077335840, 6.97486383871003407498311520985, 7.88249917501470455630746872718, 8.587342192678213596586742529397, 9.454816446534649078730744783040
