L(s) = 1 | − 2.35·2-s + 1.78·3-s + 3.54·4-s + 3.23·5-s − 4.20·6-s + 7-s − 3.63·8-s + 0.189·9-s − 7.61·10-s − 2.45·11-s + 6.33·12-s − 2.30·13-s − 2.35·14-s + 5.77·15-s + 1.47·16-s − 3.53·17-s − 0.446·18-s − 8.22·19-s + 11.4·20-s + 1.78·21-s + 5.77·22-s − 6.49·24-s + 5.45·25-s + 5.43·26-s − 5.01·27-s + 3.54·28-s + 5.84·29-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 1.03·3-s + 1.77·4-s + 1.44·5-s − 1.71·6-s + 0.377·7-s − 1.28·8-s + 0.0632·9-s − 2.40·10-s − 0.739·11-s + 1.82·12-s − 0.640·13-s − 0.629·14-s + 1.49·15-s + 0.368·16-s − 0.856·17-s − 0.105·18-s − 1.88·19-s + 2.56·20-s + 0.389·21-s + 1.23·22-s − 1.32·24-s + 1.09·25-s + 1.06·26-s − 0.965·27-s + 0.669·28-s + 1.08·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 1.78T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 2.45T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 + 3.53T + 17T^{2} \) |
| 19 | \( 1 + 8.22T + 19T^{2} \) |
| 29 | \( 1 - 5.84T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 4.08T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 5.11T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 3.00T + 61T^{2} \) |
| 67 | \( 1 - 4.21T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 0.503T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 7.22T + 83T^{2} \) |
| 89 | \( 1 + 6.03T + 89T^{2} \) |
| 97 | \( 1 + 7.08T + 97T^{2} \) |
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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
−8.422141996042687996647034979893, −7.71812964659663309151013909033, −6.97863658896052218858964096598, −6.20311034472928676090203703354, −5.34596423882197520029835642619, −4.23446046361316480811470365672, −2.65737837131466709947011140010, −2.29863069604812072933607049662, −1.66010976000980850011095387964, 0,
1.66010976000980850011095387964, 2.29863069604812072933607049662, 2.65737837131466709947011140010, 4.23446046361316480811470365672, 5.34596423882197520029835642619, 6.20311034472928676090203703354, 6.97863658896052218858964096598, 7.71812964659663309151013909033, 8.422141996042687996647034979893