L(s) = 1 | + 0.765·2-s − 3-s − 0.414·4-s − 1.84·5-s − 0.765·6-s − 1.08·8-s + 9-s − 1.41·10-s − 0.765·11-s + 0.414·12-s + 13-s + 1.84·15-s − 0.414·16-s + 0.765·18-s + 0.765·20-s − 0.585·22-s + 1.08·24-s + 2.41·25-s + 0.765·26-s − 27-s + 1.41·30-s + 0.765·32-s + 0.765·33-s − 0.414·36-s − 39-s + 2·40-s + 1.84·41-s + ⋯ |
L(s) = 1 | + 0.765·2-s − 3-s − 0.414·4-s − 1.84·5-s − 0.765·6-s − 1.08·8-s + 9-s − 1.41·10-s − 0.765·11-s + 0.414·12-s + 13-s + 1.84·15-s − 0.414·16-s + 0.765·18-s + 0.765·20-s − 0.585·22-s + 1.08·24-s + 2.41·25-s + 0.765·26-s − 27-s + 1.41·30-s + 0.765·32-s + 0.765·33-s − 0.414·36-s − 39-s + 2·40-s + 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6012364200\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6012364200\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 0.765T + T^{2} \) |
| 5 | \( 1 + 1.84T + T^{2} \) |
| 11 | \( 1 + 0.765T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.84T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.84T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + 0.765T + T^{2} \) |
| 89 | \( 1 + 0.765T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.324733425137746576434935613358, −8.469831033372964199310140398102, −7.74900896164872676274169349987, −6.99103303594728810956426898496, −5.97274546534083516325077659430, −5.28458378682711212309193493770, −4.27886054153433148109746756755, −4.03489771838745597915583045712, −2.96659640204704694745021215345, −0.71873703260887446833908055288,
0.71873703260887446833908055288, 2.96659640204704694745021215345, 4.03489771838745597915583045712, 4.27886054153433148109746756755, 5.28458378682711212309193493770, 5.97274546534083516325077659430, 6.99103303594728810956426898496, 7.74900896164872676274169349987, 8.469831033372964199310140398102, 9.324733425137746576434935613358