| L(s) = 1 | − 1.80·3-s − 4-s − 1.24i·5-s + 2.24·9-s + i·11-s + 1.80·12-s + 2.24i·15-s + 16-s + 1.24i·20-s + 0.445·23-s − 0.554·25-s − 2.24·27-s + 0.445i·31-s − 1.80i·33-s − 2.24·36-s − 1.80i·37-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 4-s − 1.24i·5-s + 2.24·9-s + i·11-s + 1.80·12-s + 2.24i·15-s + 16-s + 1.24i·20-s + 0.445·23-s − 0.554·25-s − 2.24·27-s + 0.445i·31-s − 1.80i·33-s − 2.24·36-s − 1.80i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0304 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0304 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3951374130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3951374130\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T^{2} \) |
| 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 + 1.24iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 0.445T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.445iT - T^{2} \) |
| 37 | \( 1 + 1.80iT - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 0.445iT - T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 + 1.80iT - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 2iT - T^{2} \) |
| 71 | \( 1 - 1.80iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 0.445iT - T^{2} \) |
| 97 | \( 1 + 1.24iT - T^{2} \) |
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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
−9.390022355540685296259342485331, −8.582514015945844505104488404795, −7.57872203961284111864179366433, −6.72493464585058008276502127904, −5.68233210987615756807455568522, −5.11626234604036791012327978329, −4.64727314101231391442738828658, −3.89095650552694975073236874787, −1.62570611372169132636489885406, −0.50609440589796301196721810444,
1.05586440632393924608716044149, 2.99796074628057201965664008900, 4.01779750961248794359698299071, 4.91060803514683649401916423013, 5.69155383273447227224494812507, 6.30281733290374399460406764833, 7.00299668404686698703267398455, 7.927723737842986101778556541009, 8.957881883465966454544097408150, 10.00114449783916871079011805626
