| L(s) = 1 | + 3-s − 5-s + 9-s − 13-s − 15-s + 17-s − 4·23-s + 25-s + 27-s − 6·29-s + 8·31-s − 6·37-s − 39-s − 6·41-s − 4·43-s − 45-s + 8·47-s − 7·49-s + 51-s − 2·53-s + 12·59-s − 2·61-s + 65-s − 8·67-s − 4·69-s − 8·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.140·51-s − 0.274·53-s + 1.56·59-s − 0.256·61-s + 0.124·65-s − 0.977·67-s − 0.481·69-s − 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.22093103345063, −12.88551740567249, −12.18764419730270, −11.88872591346626, −11.53080128905199, −10.87364441618636, −10.28245728081612, −10.00480626256920, −9.538628331869826, −8.883166771957594, −8.428245558757113, −8.188282438661947, −7.449360639427611, −7.199085361942285, −6.661778693373221, −5.930180879767985, −5.590454760105122, −4.773456727649194, −4.441529338149161, −3.844367927078155, −3.232819453210594, −2.926123864803531, −2.015056322982425, −1.698575608872260, −0.7675444644609389, 0,
0.7675444644609389, 1.698575608872260, 2.015056322982425, 2.926123864803531, 3.232819453210594, 3.844367927078155, 4.441529338149161, 4.773456727649194, 5.590454760105122, 5.930180879767985, 6.661778693373221, 7.199085361942285, 7.449360639427611, 8.188282438661947, 8.428245558757113, 8.883166771957594, 9.538628331869826, 10.00480626256920, 10.28245728081612, 10.87364441618636, 11.53080128905199, 11.88872591346626, 12.18764419730270, 12.88551740567249, 13.22093103345063
