| L(s) = 1 | − 4-s + 2·5-s + 2·9-s − 6·11-s + 16-s + 2·19-s − 2·20-s − 25-s + 2·29-s − 12·31-s − 2·36-s + 5·41-s + 6·44-s + 4·45-s − 6·49-s − 12·55-s + 4·59-s − 8·61-s − 64-s − 2·76-s + 12·79-s + 2·80-s − 5·81-s + 24·89-s + 4·95-s − 12·99-s + 100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s + 2/3·9-s − 1.80·11-s + 1/4·16-s + 0.458·19-s − 0.447·20-s − 1/5·25-s + 0.371·29-s − 2.15·31-s − 1/3·36-s + 0.780·41-s + 0.904·44-s + 0.596·45-s − 6/7·49-s − 1.61·55-s + 0.520·59-s − 1.02·61-s − 1/8·64-s − 0.229·76-s + 1.35·79-s + 0.223·80-s − 5/9·81-s + 2.54·89-s + 0.410·95-s − 1.20·99-s + 1/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7902602434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7902602434\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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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
−12.76805004435645432862239245801, −12.01358658782348319686332426916, −11.12838465475226465031138874472, −10.50181056919323422276502130066, −10.19524677105078893381228465769, −9.400484056125080834230342955772, −9.109403373605692684727508650678, −7.988358683254346292603082190295, −7.67909308622040811984694105441, −6.80904110843827372632476648277, −5.78628155331956279239987813546, −5.33426293676225140059122001280, −4.55099246561492137562690343487, −3.34495107937233360156966555355, −2.09231570919707459296302770195,
2.09231570919707459296302770195, 3.34495107937233360156966555355, 4.55099246561492137562690343487, 5.33426293676225140059122001280, 5.78628155331956279239987813546, 6.80904110843827372632476648277, 7.67909308622040811984694105441, 7.988358683254346292603082190295, 9.109403373605692684727508650678, 9.400484056125080834230342955772, 10.19524677105078893381228465769, 10.50181056919323422276502130066, 11.12838465475226465031138874472, 12.01358658782348319686332426916, 12.76805004435645432862239245801
