| L(s) = 1 | − 0.417·2-s − 1.82·4-s + 2.09·5-s + 0.600·7-s + 1.59·8-s − 0.876·10-s + 1.33·11-s − 5.18·13-s − 0.250·14-s + 2.98·16-s − 1.50·19-s − 3.83·20-s − 0.557·22-s + 3.77·23-s − 0.590·25-s + 2.16·26-s − 1.09·28-s + 3.55·29-s − 4.74·31-s − 4.44·32-s + 1.26·35-s + 2.84·37-s + 0.630·38-s + 3.35·40-s + 3.52·41-s − 5.10·43-s − 2.43·44-s + ⋯ |
| L(s) = 1 | − 0.295·2-s − 0.912·4-s + 0.939·5-s + 0.226·7-s + 0.564·8-s − 0.277·10-s + 0.402·11-s − 1.43·13-s − 0.0670·14-s + 0.746·16-s − 0.346·19-s − 0.857·20-s − 0.118·22-s + 0.787·23-s − 0.118·25-s + 0.424·26-s − 0.207·28-s + 0.659·29-s − 0.853·31-s − 0.784·32-s + 0.213·35-s + 0.468·37-s + 0.102·38-s + 0.530·40-s + 0.550·41-s − 0.778·43-s − 0.367·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.417T + 2T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 7 | \( 1 - 0.600T + 7T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 13 | \( 1 + 5.18T + 13T^{2} \) |
| 19 | \( 1 + 1.50T + 19T^{2} \) |
| 23 | \( 1 - 3.77T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 - 2.84T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 5.10T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 - 0.0714T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 7.13T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.61916184681479140188215273670, −6.85806277816430042068557033018, −6.08605295288977587562855132562, −5.25459681475748349906626156423, −4.82455083707384140455157517860, −4.06741810712500574372796818510, −3.02975382388875612089916254469, −2.10717631948407697402955060341, −1.24657621208496626303171251754, 0,
1.24657621208496626303171251754, 2.10717631948407697402955060341, 3.02975382388875612089916254469, 4.06741810712500574372796818510, 4.82455083707384140455157517860, 5.25459681475748349906626156423, 6.08605295288977587562855132562, 6.85806277816430042068557033018, 7.61916184681479140188215273670
