L(s) = 1 | − 3-s + 4.18·5-s + 7-s + 9-s + 11-s − 3.17·13-s − 4.18·15-s + 6.85·17-s + 0.318·19-s − 21-s + 1.87·23-s + 12.5·25-s − 27-s − 3.17·29-s − 9.23·31-s − 33-s + 4.18·35-s − 7.55·37-s + 3.17·39-s + 9.36·41-s + 10.8·43-s + 4.18·45-s + 8.06·47-s + 49-s − 6.85·51-s + 0.508·53-s + 4.18·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.87·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s − 0.880·13-s − 1.08·15-s + 1.66·17-s + 0.0731·19-s − 0.218·21-s + 0.390·23-s + 2.51·25-s − 0.192·27-s − 0.589·29-s − 1.65·31-s − 0.174·33-s + 0.708·35-s − 1.24·37-s + 0.508·39-s + 1.46·41-s + 1.66·43-s + 0.624·45-s + 1.17·47-s + 0.142·49-s − 0.959·51-s + 0.0698·53-s + 0.564·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.551942763\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.551942763\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 4.18T + 5T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 - 0.318T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 9.23T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 - 9.36T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 - 0.508T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2.66T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 + 5.01T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 + 1.74T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.765666795221537768398354932273, −7.49788177438515530215528087373, −7.09734673673948333306587917000, −6.01932050886035694863227706430, −5.52194199390554677415861790925, −5.15921322296804708298016368780, −3.97929745791476547864284670174, −2.75868645761216958515863969917, −1.89182752987041465621278356610, −1.03220033152491000292246679950,
1.03220033152491000292246679950, 1.89182752987041465621278356610, 2.75868645761216958515863969917, 3.97929745791476547864284670174, 5.15921322296804708298016368780, 5.52194199390554677415861790925, 6.01932050886035694863227706430, 7.09734673673948333306587917000, 7.49788177438515530215528087373, 8.765666795221537768398354932273