L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 7-s − 2·9-s − 3·11-s + 2·12-s + 13-s − 2·14-s − 4·16-s + 7·17-s − 4·18-s − 21-s − 6·22-s + 6·23-s + 2·26-s − 5·27-s − 2·28-s − 5·29-s + 2·31-s − 8·32-s − 3·33-s + 14·34-s − 4·36-s + 2·37-s + 39-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s + 0.277·13-s − 0.534·14-s − 16-s + 1.69·17-s − 0.942·18-s − 0.218·21-s − 1.27·22-s + 1.25·23-s + 0.392·26-s − 0.962·27-s − 0.377·28-s − 0.928·29-s + 0.359·31-s − 1.41·32-s − 0.522·33-s + 2.40·34-s − 2/3·36-s + 0.328·37-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.347442534\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.347442534\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.01254965419346541735155009772, −12.04804768145326897305333226930, −11.05181111943764343536756650509, −9.734749354117359778149997929022, −8.569343753087953468815992872802, −7.38451703018051703490900915946, −5.93898740434763322465740513982, −5.14936161735373762545181523253, −3.57752582126466520607951640819, −2.76472422088648867371864611007,
2.76472422088648867371864611007, 3.57752582126466520607951640819, 5.14936161735373762545181523253, 5.93898740434763322465740513982, 7.38451703018051703490900915946, 8.569343753087953468815992872802, 9.734749354117359778149997929022, 11.05181111943764343536756650509, 12.04804768145326897305333226930, 13.01254965419346541735155009772