L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 3·11-s − 12-s − 5·13-s + 16-s + 18-s + 5·19-s + 3·22-s + 9·23-s − 24-s − 5·26-s − 27-s − 10·31-s + 32-s − 3·33-s + 36-s + 37-s + 5·38-s + 5·39-s + 9·41-s − 8·43-s + 3·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 1.38·13-s + 1/4·16-s + 0.235·18-s + 1.14·19-s + 0.639·22-s + 1.87·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 1.79·31-s + 0.176·32-s − 0.522·33-s + 1/6·36-s + 0.164·37-s + 0.811·38-s + 0.800·39-s + 1.40·41-s − 1.21·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.772411242\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.772411242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.38318311575492912580752086205, −7.24537306100657345743138474277, −6.52776248037372836165368712817, −5.56965677904299709264619936739, −5.17007363689409040864386974659, −4.47661302612181224325724331948, −3.61431137076702225091678125525, −2.86246235895576498082479064414, −1.82973924538159961578654139058, −0.792180125052384567724417088157,
0.792180125052384567724417088157, 1.82973924538159961578654139058, 2.86246235895576498082479064414, 3.61431137076702225091678125525, 4.47661302612181224325724331948, 5.17007363689409040864386974659, 5.56965677904299709264619936739, 6.52776248037372836165368712817, 7.24537306100657345743138474277, 7.38318311575492912580752086205