| L(s) = 1 | + 4·7-s + 6·9-s + 12·17-s − 16·23-s + 6·25-s + 20·31-s + 12·41-s − 4·47-s − 2·49-s + 24·63-s − 20·71-s − 4·73-s − 8·79-s + 27·81-s + 12·89-s + 4·97-s + 16·103-s + 28·113-s + 48·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 72·153-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2·9-s + 2.91·17-s − 3.33·23-s + 6/5·25-s + 3.59·31-s + 1.87·41-s − 0.583·47-s − 2/7·49-s + 3.02·63-s − 2.37·71-s − 0.468·73-s − 0.900·79-s + 3·81-s + 1.27·89-s + 0.406·97-s + 1.57·103-s + 2.63·113-s + 4.40·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 5.82·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.718734848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.718734848\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.511218615322201476125324886202, −8.435339999531146319969007556001, −7.85052329033276788798317621047, −7.77674216922836956532674511703, −7.51739694025035837376805256081, −7.20964060717829026776974784389, −6.27766473415904451839885481166, −6.27567747924984960660861854308, −5.97648036585378566945259649141, −5.27643298662796075001281096075, −4.80514747541410603419948792692, −4.68164800387422122711270661767, −4.13920696068713736577114544821, −3.99040548654153492098683079604, −3.24384143830868426317446865170, −2.85609335182729288188206440351, −2.10261292904118788852418234774, −1.67257766671392312969540112048, −1.14906112456943752191491702303, −0.894354999349218799491935799062,
0.894354999349218799491935799062, 1.14906112456943752191491702303, 1.67257766671392312969540112048, 2.10261292904118788852418234774, 2.85609335182729288188206440351, 3.24384143830868426317446865170, 3.99040548654153492098683079604, 4.13920696068713736577114544821, 4.68164800387422122711270661767, 4.80514747541410603419948792692, 5.27643298662796075001281096075, 5.97648036585378566945259649141, 6.27567747924984960660861854308, 6.27766473415904451839885481166, 7.20964060717829026776974784389, 7.51739694025035837376805256081, 7.77674216922836956532674511703, 7.85052329033276788798317621047, 8.435339999531146319969007556001, 8.511218615322201476125324886202
