L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 12-s − 13-s − 15-s + 16-s + 18-s − 20-s + 24-s + 25-s − 26-s + 27-s − 30-s + 32-s + 36-s − 39-s − 40-s − 2·41-s − 2·43-s − 45-s + 48-s + ⋯ |
L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 12-s − 13-s − 15-s + 16-s + 18-s − 20-s + 24-s + 25-s − 26-s + 27-s − 30-s + 32-s + 36-s − 39-s − 40-s − 2·41-s − 2·43-s − 45-s + 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.436967176\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.436967176\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 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
−9.759673132189673933585306688217, −8.537353718068905978365648837870, −7.978420059161413576508647063869, −7.14942892068151733833520965737, −6.64905614040384074345212568164, −5.15016718629018384237631099509, −4.53429545289116398773298304795, −3.57911586414051145784750099109, −2.95958092461933700766334004792, −1.79523510468558883096740571807,
1.79523510468558883096740571807, 2.95958092461933700766334004792, 3.57911586414051145784750099109, 4.53429545289116398773298304795, 5.15016718629018384237631099509, 6.64905614040384074345212568164, 7.14942892068151733833520965737, 7.978420059161413576508647063869, 8.537353718068905978365648837870, 9.759673132189673933585306688217