| L(s) = 1 | − 2.74·2-s − 2.94·3-s + 5.54·4-s + 1.95·5-s + 8.09·6-s + 3.66·7-s − 9.74·8-s + 5.67·9-s − 5.36·10-s + 2.28·11-s − 16.3·12-s + 13-s − 10.0·14-s − 5.75·15-s + 15.6·16-s − 5.27·17-s − 15.5·18-s − 6.83·19-s + 10.8·20-s − 10.7·21-s − 6.27·22-s − 5.42·23-s + 28.7·24-s − 1.18·25-s − 2.74·26-s − 7.87·27-s + 20.3·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 1.70·3-s + 2.77·4-s + 0.873·5-s + 3.30·6-s + 1.38·7-s − 3.44·8-s + 1.89·9-s − 1.69·10-s + 0.689·11-s − 4.71·12-s + 0.277·13-s − 2.69·14-s − 1.48·15-s + 3.92·16-s − 1.27·17-s − 3.67·18-s − 1.56·19-s + 2.42·20-s − 2.35·21-s − 1.33·22-s − 1.13·23-s + 5.86·24-s − 0.237·25-s − 0.538·26-s − 1.51·27-s + 3.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 + 2.94T + 3T^{2} \) |
| 5 | \( 1 - 1.95T + 5T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 19 | \( 1 + 6.83T + 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 + 0.423T + 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 + 7.67T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 + 0.00498T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 - 7.35T + 67T^{2} \) |
| 71 | \( 1 - 5.73T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 2.90T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 0.448T + 97T^{2} \) |
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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
−7.51171435922627624353158613676, −6.64523028640522889435304490975, −6.36314508902080590893155830017, −5.84669543110299264643948954682, −4.89869136094887316376490428769, −4.10601664592483179582173772050, −2.25634368859287051576863981855, −1.82319957400334308550877231922, −1.05974521166093711433386861131, 0,
1.05974521166093711433386861131, 1.82319957400334308550877231922, 2.25634368859287051576863981855, 4.10601664592483179582173772050, 4.89869136094887316376490428769, 5.84669543110299264643948954682, 6.36314508902080590893155830017, 6.64523028640522889435304490975, 7.51171435922627624353158613676
