| L(s) = 1 | − 4·5-s − 9-s + 8·11-s + 12·19-s + 11·25-s + 12·29-s − 4·31-s + 24·41-s + 4·45-s − 49-s − 32·55-s − 24·59-s − 20·61-s + 16·71-s + 16·79-s + 81-s + 24·89-s − 48·95-s − 8·99-s − 20·109-s + 26·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1/3·9-s + 2.41·11-s + 2.75·19-s + 11/5·25-s + 2.22·29-s − 0.718·31-s + 3.74·41-s + 0.596·45-s − 1/7·49-s − 4.31·55-s − 3.12·59-s − 2.56·61-s + 1.89·71-s + 1.80·79-s + 1/9·81-s + 2.54·89-s − 4.92·95-s − 0.804·99-s − 1.91·109-s + 2.36·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.602249051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.602249051\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.090278864648666465341224446487, −8.218989491862241874310563928510, −8.064299758356909863675232984616, −7.55288325098702594165502106098, −7.51226570486987670801716930340, −7.10805706004313871067638185457, −6.45812107844451681419746120773, −6.26122938090300512744682722937, −6.04780182219825164866892677847, −5.23231614210226916337449333408, −4.84014988807504099471872870284, −4.56893854666655810046859401015, −4.09016668267637507937135721745, −3.55966027357382780450031188523, −3.55108049990210595287526687804, −2.89422433923791605270181446126, −2.56027479478351291638804228502, −1.33082579950994232297089636410, −1.17505176232683934023706049905, −0.61356664369414126430990298668,
0.61356664369414126430990298668, 1.17505176232683934023706049905, 1.33082579950994232297089636410, 2.56027479478351291638804228502, 2.89422433923791605270181446126, 3.55108049990210595287526687804, 3.55966027357382780450031188523, 4.09016668267637507937135721745, 4.56893854666655810046859401015, 4.84014988807504099471872870284, 5.23231614210226916337449333408, 6.04780182219825164866892677847, 6.26122938090300512744682722937, 6.45812107844451681419746120773, 7.10805706004313871067638185457, 7.51226570486987670801716930340, 7.55288325098702594165502106098, 8.064299758356909863675232984616, 8.218989491862241874310563928510, 9.090278864648666465341224446487
