L(s) = 1 | + 1.93·2-s − 3.25·3-s + 1.73·4-s + 0.302·5-s − 6.28·6-s + 2.99·7-s − 0.507·8-s + 7.57·9-s + 0.584·10-s + 5.06·11-s − 5.65·12-s + 1.94·13-s + 5.78·14-s − 0.982·15-s − 4.45·16-s − 4.04·17-s + 14.6·18-s + 4.56·19-s + 0.524·20-s − 9.72·21-s + 9.79·22-s + 5.84·23-s + 1.64·24-s − 4.90·25-s + 3.75·26-s − 14.8·27-s + 5.19·28-s + ⋯ |
L(s) = 1 | + 1.36·2-s − 1.87·3-s + 0.868·4-s + 0.135·5-s − 2.56·6-s + 1.13·7-s − 0.179·8-s + 2.52·9-s + 0.184·10-s + 1.52·11-s − 1.63·12-s + 0.538·13-s + 1.54·14-s − 0.253·15-s − 1.11·16-s − 0.981·17-s + 3.45·18-s + 1.04·19-s + 0.117·20-s − 2.12·21-s + 2.08·22-s + 1.21·23-s + 0.336·24-s − 0.981·25-s + 0.736·26-s − 2.86·27-s + 0.981·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.770323650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.770323650\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 349 | \( 1 - T \) |
good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 - 0.302T + 5T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 - 4.56T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 - 2.32T + 31T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 9.88T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 7.81T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 - 8.71T + 67T^{2} \) |
| 71 | \( 1 - 2.51T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.31T + 79T^{2} \) |
| 83 | \( 1 - 3.71T + 83T^{2} \) |
| 89 | \( 1 + 2.30T + 89T^{2} \) |
| 97 | \( 1 - 0.690T + 97T^{2} \) |
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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
−11.64604141825045233050807578147, −11.25671665325759449680651661883, −10.05388457149746096161178007697, −8.723157922907358692591500887693, −6.91220607518280401894362175659, −6.43724772134454274395093515919, −5.33146568206506353979049263127, −4.78892412031685082796261310575, −3.83626123515375622726522113693, −1.42059241554233782164567171606,
1.42059241554233782164567171606, 3.83626123515375622726522113693, 4.78892412031685082796261310575, 5.33146568206506353979049263127, 6.43724772134454274395093515919, 6.91220607518280401894362175659, 8.723157922907358692591500887693, 10.05388457149746096161178007697, 11.25671665325759449680651661883, 11.64604141825045233050807578147