| L(s) = 1 | + 2·5-s − 4·7-s + 9-s − 2·13-s − 4·23-s − 3·25-s + 2·29-s − 8·35-s + 2·45-s + 2·49-s + 6·53-s − 12·59-s − 4·63-s − 4·65-s − 8·67-s + 8·71-s − 8·81-s − 4·83-s + 8·91-s − 32·103-s + 4·107-s − 6·109-s − 8·115-s − 2·117-s + 17·121-s − 10·125-s + 127-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.834·23-s − 3/5·25-s + 0.371·29-s − 1.35·35-s + 0.298·45-s + 2/7·49-s + 0.824·53-s − 1.56·59-s − 0.503·63-s − 0.496·65-s − 0.977·67-s + 0.949·71-s − 8/9·81-s − 0.439·83-s + 0.838·91-s − 3.15·103-s + 0.386·107-s − 0.574·109-s − 0.746·115-s − 0.184·117-s + 1.54·121-s − 0.894·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 29 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 142 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.944484008314439471531146569564, −8.391059387111091082304569898852, −7.78769027397853387005458572476, −7.35557370624636616526357527142, −6.67947017652029373068355189043, −6.48488959613588788646368033785, −5.83781525425299444369164051890, −5.59611708455098243733800900727, −4.79278234445972653071474662432, −4.19128170456069501552125238629, −3.58058403021236909710312817373, −2.90774117310316373933884637970, −2.33637286847522632523061847417, −1.49599243708434003017641040934, 0,
1.49599243708434003017641040934, 2.33637286847522632523061847417, 2.90774117310316373933884637970, 3.58058403021236909710312817373, 4.19128170456069501552125238629, 4.79278234445972653071474662432, 5.59611708455098243733800900727, 5.83781525425299444369164051890, 6.48488959613588788646368033785, 6.67947017652029373068355189043, 7.35557370624636616526357527142, 7.78769027397853387005458572476, 8.391059387111091082304569898852, 8.944484008314439471531146569564
