L(s) = 1 | − 5-s − 2·7-s + 3·11-s − 13-s + 5·17-s + 6·19-s − 23-s + 25-s − 2·29-s + 11·31-s + 2·35-s − 2·37-s + 3·41-s + 43-s − 12·47-s − 3·49-s + 9·53-s − 3·55-s + 12·61-s + 65-s − 15·67-s + 12·71-s − 4·73-s − 6·77-s − 8·79-s − 3·83-s − 5·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.904·11-s − 0.277·13-s + 1.21·17-s + 1.37·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s + 1.97·31-s + 0.338·35-s − 0.328·37-s + 0.468·41-s + 0.152·43-s − 1.75·47-s − 3/7·49-s + 1.23·53-s − 0.404·55-s + 1.53·61-s + 0.124·65-s − 1.83·67-s + 1.42·71-s − 0.468·73-s − 0.683·77-s − 0.900·79-s − 0.329·83-s − 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.81707083192784, −13.25273099231686, −12.83380364521864, −12.12305106268478, −11.89950326179281, −11.58626135897282, −10.95472135516957, −10.10875833914766, −9.891963398817001, −9.601557953431061, −8.831905209159050, −8.442669299793664, −7.742421658655829, −7.404873488493944, −6.853207000487574, −6.236233085208581, −5.918385390805973, −5.069882411609143, −4.774290134100806, −3.821752331796604, −3.589419686407125, −2.981808016618459, −2.404377927454904, −1.313600263756290, −0.9710693309374795, 0,
0.9710693309374795, 1.313600263756290, 2.404377927454904, 2.981808016618459, 3.589419686407125, 3.821752331796604, 4.774290134100806, 5.069882411609143, 5.918385390805973, 6.236233085208581, 6.853207000487574, 7.404873488493944, 7.742421658655829, 8.442669299793664, 8.831905209159050, 9.601557953431061, 9.891963398817001, 10.10875833914766, 10.95472135516957, 11.58626135897282, 11.89950326179281, 12.12305106268478, 12.83380364521864, 13.25273099231686, 13.81707083192784