| L(s) = 1 | − 2.94i·3-s + i·5-s + 1.89·7-s − 5.65·9-s + 2.17·11-s − 0.0572·13-s + 2.94·15-s − 1.76i·17-s − 1.57·19-s − 5.56i·21-s + (1.82 − 4.43i)23-s − 25-s + 7.82i·27-s − 4.26·29-s − 3.54i·31-s + ⋯ |
| L(s) = 1 | − 1.69i·3-s + 0.447i·5-s + 0.714·7-s − 1.88·9-s + 0.655·11-s − 0.0158·13-s + 0.759·15-s − 0.428i·17-s − 0.361·19-s − 1.21i·21-s + (0.379 − 0.925i)23-s − 0.200·25-s + 1.50i·27-s − 0.791·29-s − 0.636i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610782994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610782994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-1.82 + 4.43i)T \) |
| good | 3 | \( 1 + 2.94iT - 3T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 + 0.0572T + 13T^{2} \) |
| 17 | \( 1 + 1.76iT - 17T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 3.54iT - 31T^{2} \) |
| 37 | \( 1 + 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 - 5.72T + 43T^{2} \) |
| 47 | \( 1 - 6.49iT - 47T^{2} \) |
| 53 | \( 1 + 11.7iT - 53T^{2} \) |
| 59 | \( 1 + 3.46iT - 59T^{2} \) |
| 61 | \( 1 + 7.00iT - 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 + 9.53iT - 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 - 2.24T + 79T^{2} \) |
| 83 | \( 1 - 0.177T + 83T^{2} \) |
| 89 | \( 1 - 10.1iT - 89T^{2} \) |
| 97 | \( 1 - 7.95iT - 97T^{2} \) |
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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
−8.766377469759179989595317027631, −7.85291890636400741179093102762, −7.45663955636748554899881650073, −6.59195933481372924804481416109, −6.07589173281172827378007195994, −5.01708173436257386624771295936, −3.79413003626275046533390184663, −2.50838871739500744067901265021, −1.83027049084395288607733441621, −0.61054635923595636901062142296,
1.48681172835085793100029961479, 3.02473756730790840488307987837, 3.97509553582873293761924460481, 4.55105846751967296343552761504, 5.30681587882614974073068949567, 6.04950912952784074789666915614, 7.30662605424576290645342411140, 8.359561911832972310825155019274, 8.881429138662886273271032960135, 9.547605405015600881604455791481
