L(s) = 1 | + 1.82·2-s − 1.95·3-s + 2.33·4-s − 3.57·6-s − 0.209·7-s + 2.44·8-s + 2.82·9-s − 4.57·12-s − 0.381·14-s + 2.12·16-s + 1.33·17-s + 5.16·18-s + 0.408·21-s − 4.78·24-s − 3.57·27-s − 0.488·28-s + 1.44·32-s + 2.44·34-s + 6.61·36-s + 0.618·37-s + 0.747·42-s + 47-s − 4.16·48-s − 0.956·49-s − 2.61·51-s − 1.61·53-s − 6.53·54-s + ⋯ |
L(s) = 1 | + 1.82·2-s − 1.95·3-s + 2.33·4-s − 3.57·6-s − 0.209·7-s + 2.44·8-s + 2.82·9-s − 4.57·12-s − 0.381·14-s + 2.12·16-s + 1.33·17-s + 5.16·18-s + 0.408·21-s − 4.78·24-s − 3.57·27-s − 0.488·28-s + 1.44·32-s + 2.44·34-s + 6.61·36-s + 0.618·37-s + 0.747·42-s + 47-s − 4.16·48-s − 0.956·49-s − 2.61·51-s − 1.61·53-s − 6.53·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.769735515\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.769735515\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 - 1.82T + T^{2} \) |
| 3 | \( 1 + 1.95T + T^{2} \) |
| 7 | \( 1 + 0.209T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.33T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 + 0.209T + T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.95T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−10.44949519627006686167851598540, −9.661991510712751037653942661973, −7.68131116424458933144929205036, −7.00990448839809895410513041694, −6.08397995550313590848118732144, −5.78608369228917528500602267950, −4.89605477301087101366288686522, −4.27824163681869166001490562653, −3.18487540244453179717487537214, −1.50002644810544973910393577969,
1.50002644810544973910393577969, 3.18487540244453179717487537214, 4.27824163681869166001490562653, 4.89605477301087101366288686522, 5.78608369228917528500602267950, 6.08397995550313590848118732144, 7.00990448839809895410513041694, 7.68131116424458933144929205036, 9.661991510712751037653942661973, 10.44949519627006686167851598540