L(s) = 1 | + 2.63·2-s − 3-s + 4.95·4-s − 0.742·5-s − 2.63·6-s + 7.80·8-s + 9-s − 1.95·10-s − 5.37·11-s − 4.95·12-s − 2.10·13-s + 0.742·15-s + 10.6·16-s − 7.85·17-s + 2.63·18-s − 7.61·19-s − 3.68·20-s − 14.1·22-s − 23-s − 7.80·24-s − 4.44·25-s − 5.54·26-s − 27-s − 2.52·29-s + 1.95·30-s + 7.75·31-s + 12.5·32-s + ⋯ |
L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.47·4-s − 0.332·5-s − 1.07·6-s + 2.75·8-s + 0.333·9-s − 0.619·10-s − 1.62·11-s − 1.43·12-s − 0.582·13-s + 0.191·15-s + 2.66·16-s − 1.90·17-s + 0.621·18-s − 1.74·19-s − 0.822·20-s − 3.02·22-s − 0.208·23-s − 1.59·24-s − 0.889·25-s − 1.08·26-s − 0.192·27-s − 0.468·29-s + 0.357·30-s + 1.39·31-s + 2.21·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 5 | \( 1 + 0.742T + 5T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 + 7.61T + 19T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 - 7.59T + 41T^{2} \) |
| 43 | \( 1 - 7.04T + 43T^{2} \) |
| 47 | \( 1 - 5.73T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 + 9.87T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 0.596T + 79T^{2} \) |
| 83 | \( 1 - 0.194T + 83T^{2} \) |
| 89 | \( 1 + 1.61T + 89T^{2} \) |
| 97 | \( 1 - 9.67T + 97T^{2} \) |
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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
−7.84030198423162685615288946026, −7.21903107912073190010168765132, −6.41123778161367642889769784664, −5.87995023718196255350016686623, −5.00809209191582257930113896059, −4.46655703407219142971292990234, −3.88802094877354311535115451669, −2.48740884125946952075983389617, −2.23791278365569032886392594517, 0,
2.23791278365569032886392594517, 2.48740884125946952075983389617, 3.88802094877354311535115451669, 4.46655703407219142971292990234, 5.00809209191582257930113896059, 5.87995023718196255350016686623, 6.41123778161367642889769784664, 7.21903107912073190010168765132, 7.84030198423162685615288946026