L(s) = 1 | + (−0.780 − 1.17i)2-s + (−0.780 + 1.84i)4-s − 1.69i·5-s + (2.56 − 0.662i)7-s + (2.78 − 0.516i)8-s + (−2 + 1.32i)10-s + 3.02i·11-s − 6.04i·13-s + (−2.78 − 2.50i)14-s + (−2.78 − 2.87i)16-s − 4.34i·17-s − 1.12·19-s + (3.12 + 1.32i)20-s + (3.56 − 2.35i)22-s − 3.02i·23-s + ⋯ |
L(s) = 1 | + (−0.552 − 0.833i)2-s + (−0.390 + 0.920i)4-s − 0.758i·5-s + (0.968 − 0.250i)7-s + (0.983 − 0.182i)8-s + (−0.632 + 0.418i)10-s + 0.910i·11-s − 1.67i·13-s + (−0.743 − 0.669i)14-s + (−0.695 − 0.718i)16-s − 1.05i·17-s − 0.257·19-s + (0.698 + 0.296i)20-s + (0.759 − 0.502i)22-s − 0.629i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.640405 - 0.743031i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.640405 - 0.743031i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.780 + 1.17i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.56 + 0.662i)T \) |
good | 5 | \( 1 + 1.69iT - 5T^{2} \) |
| 11 | \( 1 - 3.02iT - 11T^{2} \) |
| 13 | \( 1 + 6.04iT - 13T^{2} \) |
| 17 | \( 1 + 4.34iT - 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 3.02iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 7.73iT - 41T^{2} \) |
| 43 | \( 1 - 8.10iT - 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 9.43iT - 61T^{2} \) |
| 67 | \( 1 + 2.06iT - 67T^{2} \) |
| 71 | \( 1 - 12.4iT - 71T^{2} \) |
| 73 | \( 1 - 3.39iT - 73T^{2} \) |
| 79 | \( 1 + 4.71iT - 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + 7.73iT - 89T^{2} \) |
| 97 | \( 1 - 8.68iT - 97T^{2} \) |
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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
−11.79773883858831552700869990900, −10.72464191021499846125098262838, −10.02767654224515810706856063998, −8.882830805806560169550438213732, −8.096208283443308791556499903929, −7.24034433777827634857816587182, −5.18723591545151194904594203404, −4.40397294426286718002719676220, −2.70433685782308058967524019006, −1.06011230503244344915542757204,
1.86129785118577119109976243663, 4.02922784317045736189504813013, 5.39791744168898827450635389077, 6.45480972987674849815622195444, 7.31965919028364277186617675124, 8.489556537432565556908423767461, 9.068011371157277501465599856057, 10.51044086459023151239222639437, 11.03009236452252208536903888606, 12.09226187216190634196060341013