L(s) = 1 | − 1.67·2-s − 3.26·3-s + 0.821·4-s + 5.48·6-s + 7-s + 1.97·8-s + 7.64·9-s + 2.67·11-s − 2.67·12-s + 4.80·13-s − 1.67·14-s − 4.96·16-s + 3.20·17-s − 12.8·18-s + 2.76·19-s − 3.26·21-s − 4.50·22-s + 23-s − 6.46·24-s − 8.06·26-s − 15.1·27-s + 0.821·28-s − 7.60·29-s − 8.00·31-s + 4.38·32-s − 8.74·33-s − 5.38·34-s + ⋯ |
L(s) = 1 | − 1.18·2-s − 1.88·3-s + 0.410·4-s + 2.23·6-s + 0.377·7-s + 0.700·8-s + 2.54·9-s + 0.807·11-s − 0.773·12-s + 1.33·13-s − 0.448·14-s − 1.24·16-s + 0.777·17-s − 3.02·18-s + 0.633·19-s − 0.712·21-s − 0.959·22-s + 0.208·23-s − 1.31·24-s − 1.58·26-s − 2.91·27-s + 0.155·28-s − 1.41·29-s − 1.43·31-s + 0.775·32-s − 1.52·33-s − 0.923·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 3 | \( 1 + 3.26T + 3T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 - 4.80T + 13T^{2} \) |
| 17 | \( 1 - 3.20T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 + 8.00T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 8.18T + 41T^{2} \) |
| 43 | \( 1 - 5.22T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 0.178T + 53T^{2} \) |
| 59 | \( 1 + 8.96T + 59T^{2} \) |
| 61 | \( 1 - 7.14T + 61T^{2} \) |
| 67 | \( 1 + 9.06T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 + 2.44T + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 - 4.71T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 3.73T + 97T^{2} \) |
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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
−7.980024322753659999823500667465, −7.32320947418973284009820384884, −6.74283536077035856637393371484, −5.78499763275147172482646480858, −5.41054085051735988843654680815, −4.36089172186573992515905461349, −3.69040139942489194656493381627, −1.54881521268127110346560788557, −1.22517484436859893682346863186, 0,
1.22517484436859893682346863186, 1.54881521268127110346560788557, 3.69040139942489194656493381627, 4.36089172186573992515905461349, 5.41054085051735988843654680815, 5.78499763275147172482646480858, 6.74283536077035856637393371484, 7.32320947418973284009820384884, 7.980024322753659999823500667465