| L(s) = 1 | − 2.41·2-s − 1.41·3-s + 3.82·4-s + 3.41·6-s + 7-s − 4.41·8-s − 0.999·9-s + 11-s − 5.41·12-s − 0.585·13-s − 2.41·14-s + 2.99·16-s + 3.41·17-s + 2.41·18-s − 1.41·21-s − 2.41·22-s − 3.17·23-s + 6.24·24-s + 1.41·26-s + 5.65·27-s + 3.82·28-s + 4.82·29-s − 10.2·31-s + 1.58·32-s − 1.41·33-s − 8.24·34-s − 3.82·36-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.816·3-s + 1.91·4-s + 1.39·6-s + 0.377·7-s − 1.56·8-s − 0.333·9-s + 0.301·11-s − 1.56·12-s − 0.162·13-s − 0.645·14-s + 0.749·16-s + 0.828·17-s + 0.569·18-s − 0.308·21-s − 0.514·22-s − 0.661·23-s + 1.27·24-s + 0.277·26-s + 1.08·27-s + 0.723·28-s + 0.896·29-s − 1.83·31-s + 0.280·32-s − 0.246·33-s − 1.41·34-s − 0.638·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 9.07T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.637626993216522571405703844449, −8.333779797738169432481180200921, −7.29991921939543351674196313838, −6.74381544746659297900333730334, −5.77850818585989636318588950336, −5.05049025076394348387344402122, −3.60417805360382781817208872996, −2.27333022588473983824209244415, −1.20347073223442582288803737659, 0,
1.20347073223442582288803737659, 2.27333022588473983824209244415, 3.60417805360382781817208872996, 5.05049025076394348387344402122, 5.77850818585989636318588950336, 6.74381544746659297900333730334, 7.29991921939543351674196313838, 8.333779797738169432481180200921, 8.637626993216522571405703844449
