| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 2·5-s + 4·6-s + 3·7-s + 4·8-s + 3·9-s + 4·10-s + 6·12-s + 6·14-s + 4·15-s + 5·16-s + 2·17-s + 6·18-s − 3·19-s + 6·20-s + 6·21-s − 3·23-s + 8·24-s + 3·25-s + 4·27-s + 9·28-s + 9·29-s + 8·30-s − 31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 1.13·7-s + 1.41·8-s + 9-s + 1.26·10-s + 1.73·12-s + 1.60·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 0.688·19-s + 1.34·20-s + 1.30·21-s − 0.625·23-s + 1.63·24-s + 3/5·25-s + 0.769·27-s + 1.70·28-s + 1.67·29-s + 1.46·30-s − 0.179·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.46437822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.46437822\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 57 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 186 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 206 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 19 T + 244 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 144 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 186 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.294166297476729841842358491501, −8.104496163096485352392188819357, −7.73110271195495359051022879707, −7.16817843598259872212854205433, −6.89541371305685982043066593927, −6.76789969932272338760958370498, −6.03790646279397498019680485656, −5.83283435058607365525547919130, −5.32389611537743009867781153202, −5.28587988422669838596121846405, −4.54008934964114189254564718901, −4.35348328239761002176418358630, −3.85675563620953944982759087698, −3.80868281370984231879369607264, −2.90378871007548657991962624759, −2.64923246643240239205773349305, −2.23935305887453758019212679292, −2.10721198297347335620985787392, −1.14368476076863984087034754832, −1.10987255855702977992504962799,
1.10987255855702977992504962799, 1.14368476076863984087034754832, 2.10721198297347335620985787392, 2.23935305887453758019212679292, 2.64923246643240239205773349305, 2.90378871007548657991962624759, 3.80868281370984231879369607264, 3.85675563620953944982759087698, 4.35348328239761002176418358630, 4.54008934964114189254564718901, 5.28587988422669838596121846405, 5.32389611537743009867781153202, 5.83283435058607365525547919130, 6.03790646279397498019680485656, 6.76789969932272338760958370498, 6.89541371305685982043066593927, 7.16817843598259872212854205433, 7.73110271195495359051022879707, 8.104496163096485352392188819357, 8.294166297476729841842358491501
