| L(s) = 1 | − 1.53i·2-s + i·3-s − 0.369·4-s + 1.53·6-s − 4.87i·7-s − 2.51i·8-s − 9-s + 4.34·11-s − 0.369i·12-s − 2.53i·13-s − 7.51·14-s − 4.60·16-s − 2.63i·17-s + 1.53i·18-s + 7.41·19-s + ⋯ |
| L(s) = 1 | − 1.08i·2-s + 0.577i·3-s − 0.184·4-s + 0.628·6-s − 1.84i·7-s − 0.887i·8-s − 0.333·9-s + 1.30·11-s − 0.106i·12-s − 0.704i·13-s − 2.00·14-s − 1.15·16-s − 0.638i·17-s + 0.362i·18-s + 1.70·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.017840269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.017840269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53iT - 2T^{2} \) |
| 7 | \( 1 + 4.87iT - 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 + 2.53iT - 13T^{2} \) |
| 17 | \( 1 + 2.63iT - 17T^{2} \) |
| 19 | \( 1 - 7.41T + 19T^{2} \) |
| 23 | \( 1 - 2.29iT - 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 37 | \( 1 - 5.80iT - 37T^{2} \) |
| 41 | \( 1 + 0.183T + 41T^{2} \) |
| 43 | \( 1 + 6.49iT - 43T^{2} \) |
| 47 | \( 1 - 9.80iT - 47T^{2} \) |
| 53 | \( 1 + 1.86iT - 53T^{2} \) |
| 59 | \( 1 - 7.90T + 59T^{2} \) |
| 61 | \( 1 + 8.15T + 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 + 15.5iT - 73T^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 - 6.38iT - 83T^{2} \) |
| 89 | \( 1 - 7.51T + 89T^{2} \) |
| 97 | \( 1 + 16.2iT - 97T^{2} \) |
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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
−9.176077369077771661000500692360, −7.73422003807179847028719287006, −7.24571912665951172437821319961, −6.44320885732995043874130185228, −5.23439181971456432250952976011, −4.24585045181561266654510567107, −3.61534898817503106450859717407, −3.09125648967402411193862923194, −1.48893797965191274452856870840, −0.71930086963454833009568608178,
1.63780681937313037837742033436, 2.41790377042869799136558479240, 3.61935233277667443273047991608, 5.00329301722710883398459124909, 5.69894898713347118569923387954, 6.23691087984381599764973365471, 6.90146689872010760602347528310, 7.68367395360029438921421137335, 8.491463710699502466267057225491, 9.086012098966214386251705372406
