| L(s) = 1 | − 3·3-s + 5-s + 6·9-s − 11-s − 3·13-s − 3·15-s − 3·17-s − 6·19-s − 4·23-s + 25-s − 9·27-s + 29-s + 6·31-s + 3·33-s + 9·39-s + 6·41-s + 6·43-s + 6·45-s − 9·47-s + 9·51-s + 10·53-s − 55-s + 18·57-s + 6·59-s − 3·65-s + 14·67-s + 12·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s − 0.301·11-s − 0.832·13-s − 0.774·15-s − 0.727·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s − 1.73·27-s + 0.185·29-s + 1.07·31-s + 0.522·33-s + 1.44·39-s + 0.937·41-s + 0.914·43-s + 0.894·45-s − 1.31·47-s + 1.26·51-s + 1.37·53-s − 0.134·55-s + 2.38·57-s + 0.781·59-s − 0.372·65-s + 1.71·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−16.34597150818796, −15.88237434301519, −15.32847239879278, −14.67963337058737, −14.02527607274658, −13.26407624945671, −12.75209402853527, −12.40685942332952, −11.70010928197103, −11.26159180304908, −10.66733668918765, −10.11271575313026, −9.820214470817458, −8.887946965594850, −8.192039013172773, −7.387802858599121, −6.720167889931798, −6.293642099924552, −5.745791152250997, −5.067438592249797, −4.514321039088832, −3.976293979182003, −2.575651535825041, −1.994159049888275, −0.8475101528438349, 0,
0.8475101528438349, 1.994159049888275, 2.575651535825041, 3.976293979182003, 4.514321039088832, 5.067438592249797, 5.745791152250997, 6.293642099924552, 6.720167889931798, 7.387802858599121, 8.192039013172773, 8.887946965594850, 9.820214470817458, 10.11271575313026, 10.66733668918765, 11.26159180304908, 11.70010928197103, 12.40685942332952, 12.75209402853527, 13.26407624945671, 14.02527607274658, 14.67963337058737, 15.32847239879278, 15.88237434301519, 16.34597150818796
