L(s) = 1 | − 3-s − 2·7-s + 9-s + 4·11-s − 2·13-s + 4·19-s + 2·21-s + 6·23-s − 27-s − 29-s − 8·31-s − 4·33-s + 4·37-s + 2·39-s + 6·41-s − 8·43-s + 8·47-s − 3·49-s + 2·53-s − 4·57-s + 10·61-s − 2·63-s − 2·67-s − 6·69-s + 8·71-s − 4·73-s − 8·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.917·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s − 0.185·29-s − 1.43·31-s − 0.696·33-s + 0.657·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.529·57-s + 1.28·61-s − 0.251·63-s − 0.244·67-s − 0.722·69-s + 0.949·71-s − 0.468·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.503331623\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.503331623\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.43155251618781068734303576388, −7.08699430241840280426684698717, −6.43905039124362280609784468854, −5.69797450406483927177719556955, −5.11203831078664002015105201377, −4.21203517302424004731401490495, −3.53976532262690206076786897710, −2.73583253298118359397706904438, −1.57088645165968360850282090543, −0.64141448637663671104814541446,
0.64141448637663671104814541446, 1.57088645165968360850282090543, 2.73583253298118359397706904438, 3.53976532262690206076786897710, 4.21203517302424004731401490495, 5.11203831078664002015105201377, 5.69797450406483927177719556955, 6.43905039124362280609784468854, 7.08699430241840280426684698717, 7.43155251618781068734303576388