| 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
| \(L(1)\) |
\(\approx\) |
\(2.525825723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.525825723\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 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$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 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$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−8.882973390919431189931546030023, −8.667294171962000083603834753027, −8.126447405259108290149301827877, −7.55374057024406981139884461541, −7.26722305485733830435611542475, −6.64616217711764103051059194594, −6.08435316938499740859909385468, −5.48768369864301108603719423389, −5.10788799131199125481985912410, −4.63621825783584592948363037107, −4.08217090333845652033205832309, −3.26327287976641646801147029720, −2.35779664044646466612681659878, −1.93007995861087487258920319346, −1.09544774860714708023912301414,
1.09544774860714708023912301414, 1.93007995861087487258920319346, 2.35779664044646466612681659878, 3.26327287976641646801147029720, 4.08217090333845652033205832309, 4.63621825783584592948363037107, 5.10788799131199125481985912410, 5.48768369864301108603719423389, 6.08435316938499740859909385468, 6.64616217711764103051059194594, 7.26722305485733830435611542475, 7.55374057024406981139884461541, 8.126447405259108290149301827877, 8.667294171962000083603834753027, 8.882973390919431189931546030023
