| L(s) = 1 | − 0.937i·2-s + 1.03i·3-s + 1.12·4-s + (−1.84 − 1.26i)5-s + 0.965·6-s − 1.65i·7-s − 2.92i·8-s + 1.93·9-s + (−1.18 + 1.72i)10-s + 4.63·11-s + 1.15i·12-s + 5.55i·13-s − 1.55·14-s + (1.30 − 1.90i)15-s − 0.497·16-s + 7.23i·17-s + ⋯ |
| L(s) = 1 | − 0.662i·2-s + 0.594i·3-s + 0.560·4-s + (−0.824 − 0.565i)5-s + 0.394·6-s − 0.626i·7-s − 1.03i·8-s + 0.646·9-s + (−0.374 + 0.546i)10-s + 1.39·11-s + 0.333i·12-s + 1.54i·13-s − 0.415·14-s + (0.336 − 0.490i)15-s − 0.124·16-s + 1.75i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.142721536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142721536\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.84 + 1.26i)T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.937iT - 2T^{2} \) |
| 3 | \( 1 - 1.03iT - 3T^{2} \) |
| 7 | \( 1 + 1.65iT - 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 - 5.55iT - 13T^{2} \) |
| 17 | \( 1 - 7.23iT - 17T^{2} \) |
| 23 | \( 1 + 4.65iT - 23T^{2} \) |
| 29 | \( 1 + 5.69T + 29T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 + 0.159iT - 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 + 3.60iT - 43T^{2} \) |
| 47 | \( 1 + 7.34iT - 47T^{2} \) |
| 53 | \( 1 + 0.175iT - 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 - 6.22T + 61T^{2} \) |
| 67 | \( 1 + 1.45iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 2.72iT - 73T^{2} \) |
| 79 | \( 1 + 2.47T + 79T^{2} \) |
| 83 | \( 1 + 5.13iT - 83T^{2} \) |
| 89 | \( 1 - 3.66T + 89T^{2} \) |
| 97 | \( 1 - 5.45iT - 97T^{2} \) |
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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
−9.248264784617904255306370933737, −8.684538706540986784916894939389, −7.52712169775847581258355823562, −6.83425265102113551724801109779, −6.18823009174318351885113482337, −4.55162699987385134200986864689, −3.96598812502222214807269273885, −3.72619844289597477715664371378, −1.93905673873492104704014120719, −1.10766854216506829869853236237,
1.04221071711508462074002121182, 2.48086054064100701316440466044, 3.26210959296877265934326222140, 4.49233692586352936291815899124, 5.62209886550445843596473280398, 6.31337519544654751960513439926, 7.17804857580034374384893669891, 7.48729487435821395716362650530, 8.197530373332173171777676898924, 9.208719407262711906187644756593
