L(s) = 1 | − 2·5-s − 4·7-s − 3·9-s − 13-s − 6·17-s + 3·19-s − 9·23-s − 25-s − 10·29-s − 8·31-s + 8·35-s + 11·37-s − 4·43-s + 6·45-s − 3·47-s + 9·49-s − 53-s − 13·59-s + 3·61-s + 12·63-s + 2·65-s + 8·67-s − 8·71-s − 4·73-s + 6·79-s + 9·81-s + 6·83-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 1.51·7-s − 9-s − 0.277·13-s − 1.45·17-s + 0.688·19-s − 1.87·23-s − 1/5·25-s − 1.85·29-s − 1.43·31-s + 1.35·35-s + 1.80·37-s − 0.609·43-s + 0.894·45-s − 0.437·47-s + 9/7·49-s − 0.137·53-s − 1.69·59-s + 0.384·61-s + 1.51·63-s + 0.248·65-s + 0.977·67-s − 0.949·71-s − 0.468·73-s + 0.675·79-s + 81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.52542124636976209696731698498, −6.56870323779876583978884872592, −6.05066158186841973885179566530, −5.34614084489609530249489198255, −4.14580059041744850412142656372, −3.70682191769348489151611267627, −2.90206741182725093286474673544, −2.02655141313196498091509295261, 0, 0,
2.02655141313196498091509295261, 2.90206741182725093286474673544, 3.70682191769348489151611267627, 4.14580059041744850412142656372, 5.34614084489609530249489198255, 6.05066158186841973885179566530, 6.56870323779876583978884872592, 7.52542124636976209696731698498