| L(s) = 1 | − 2.65·2-s − 3-s + 5.03·4-s − 3.48·5-s + 2.65·6-s − 7-s − 8.03·8-s + 9-s + 9.25·10-s − 1.94·11-s − 5.03·12-s + 13-s + 2.65·14-s + 3.48·15-s + 11.2·16-s − 17-s − 2.65·18-s + 4.49·19-s − 17.5·20-s + 21-s + 5.16·22-s − 4.84·23-s + 8.03·24-s + 7.17·25-s − 2.65·26-s − 27-s − 5.03·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.51·4-s − 1.56·5-s + 1.08·6-s − 0.377·7-s − 2.84·8-s + 0.333·9-s + 2.92·10-s − 0.586·11-s − 1.45·12-s + 0.277·13-s + 0.708·14-s + 0.900·15-s + 2.81·16-s − 0.242·17-s − 0.624·18-s + 1.03·19-s − 3.92·20-s + 0.218·21-s + 1.10·22-s − 1.01·23-s + 1.64·24-s + 1.43·25-s − 0.520·26-s − 0.192·27-s − 0.950·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1854389030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1854389030\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 11 | \( 1 + 1.94T + 11T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 + 4.84T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 - 0.935T + 43T^{2} \) |
| 47 | \( 1 + 9.60T + 47T^{2} \) |
| 53 | \( 1 + 3.76T + 53T^{2} \) |
| 59 | \( 1 - 8.79T + 59T^{2} \) |
| 61 | \( 1 + 4.75T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 + 8.34T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 - 1.45T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.185553795893654533774357524412, −7.82045708435534746318411773687, −7.11361154742090092431566036199, −6.56990818163836860240098870263, −5.67477786225848154080239937099, −4.56287131195753841528561865641, −3.50203479444732149404482024112, −2.72061736086303526410902916387, −1.38086337888323543488504193088, −0.34726065561789998709713469791,
0.34726065561789998709713469791, 1.38086337888323543488504193088, 2.72061736086303526410902916387, 3.50203479444732149404482024112, 4.56287131195753841528561865641, 5.67477786225848154080239937099, 6.56990818163836860240098870263, 7.11361154742090092431566036199, 7.82045708435534746318411773687, 8.185553795893654533774357524412
