L(s) = 1 | + 2.38·2-s − 1.09·3-s + 3.70·4-s − 2.61·6-s − 7-s + 4.05·8-s − 1.79·9-s + 0.391·11-s − 4.05·12-s − 4.90·13-s − 2.38·14-s + 2.29·16-s + 4.88·17-s − 4.29·18-s + 8.50·19-s + 1.09·21-s + 0.935·22-s − 23-s − 4.44·24-s − 11.7·26-s + 5.25·27-s − 3.70·28-s + 4.87·29-s + 6.31·31-s − 2.64·32-s − 0.429·33-s + 11.6·34-s + ⋯ |
L(s) = 1 | + 1.68·2-s − 0.632·3-s + 1.85·4-s − 1.06·6-s − 0.377·7-s + 1.43·8-s − 0.599·9-s + 0.118·11-s − 1.17·12-s − 1.36·13-s − 0.638·14-s + 0.573·16-s + 1.18·17-s − 1.01·18-s + 1.95·19-s + 0.239·21-s + 0.199·22-s − 0.208·23-s − 0.907·24-s − 2.29·26-s + 1.01·27-s − 0.699·28-s + 0.904·29-s + 1.13·31-s − 0.467·32-s − 0.0747·33-s + 1.99·34-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
\(L(1)\) |
\(\approx\) |
\(3.898448156\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.898448156\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.38T + 2T^{2} \) |
| 3 | \( 1 + 1.09T + 3T^{2} \) |
| 11 | \( 1 - 0.391T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 19 | \( 1 - 8.50T + 19T^{2} \) |
| 29 | \( 1 - 4.87T + 29T^{2} \) |
| 31 | \( 1 - 6.31T + 31T^{2} \) |
| 37 | \( 1 - 8.31T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 + 2.18T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 - 7.03T + 53T^{2} \) |
| 59 | \( 1 + 9.90T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 5.94T + 67T^{2} \) |
| 71 | \( 1 + 3.44T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 - 7.83T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 - 7.33T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.108529407137645260254337381594, −7.41777344434740541067464733158, −6.64489253125624901474878557296, −5.99542942461168026698641045997, −5.26529925597935629643773174568, −4.96929036072563860609830728180, −3.94318481418851208165038456415, −2.99955569560324949065783972585, −2.59109990806751942198462602042, −0.906143181467533473667367321976,
0.906143181467533473667367321976, 2.59109990806751942198462602042, 2.99955569560324949065783972585, 3.94318481418851208165038456415, 4.96929036072563860609830728180, 5.26529925597935629643773174568, 5.99542942461168026698641045997, 6.64489253125624901474878557296, 7.41777344434740541067464733158, 8.108529407137645260254337381594