| L(s) = 1 | − 2·7-s + 2·11-s − 4·13-s − 2·17-s + 19-s − 4·23-s + 8·31-s − 8·37-s + 8·41-s − 6·43-s + 12·47-s − 3·49-s − 6·53-s + 2·61-s + 8·67-s − 8·71-s − 14·73-s − 4·77-s − 4·83-s + 8·91-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 4·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.603·11-s − 1.10·13-s − 0.485·17-s + 0.229·19-s − 0.834·23-s + 1.43·31-s − 1.31·37-s + 1.24·41-s − 0.914·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.256·61-s + 0.977·67-s − 0.949·71-s − 1.63·73-s − 0.455·77-s − 0.439·83-s + 0.838·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.366·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.156743936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156743936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.16415462272006, −13.77504859903541, −13.07107065327284, −12.69100865329668, −12.08081263792771, −11.81555216362233, −11.24258580519524, −10.47137502245474, −10.01286871725969, −9.763796739508145, −8.957258998362094, −8.767017880704897, −7.892385509521041, −7.439700556808173, −6.868086991496760, −6.354661059087883, −5.912073794291847, −5.155430896568967, −4.570556240248185, −4.040050585986159, −3.361655489681421, −2.715355193551842, −2.158420166850931, −1.303468442048136, −0.3641827298438775,
0.3641827298438775, 1.303468442048136, 2.158420166850931, 2.715355193551842, 3.361655489681421, 4.040050585986159, 4.570556240248185, 5.155430896568967, 5.912073794291847, 6.354661059087883, 6.868086991496760, 7.439700556808173, 7.892385509521041, 8.767017880704897, 8.957258998362094, 9.763796739508145, 10.01286871725969, 10.47137502245474, 11.24258580519524, 11.81555216362233, 12.08081263792771, 12.69100865329668, 13.07107065327284, 13.77504859903541, 14.16415462272006
